Symmetry in the Simultaneity of events in Special Relativity I have been bothered for long by this question and finally decided to ask it here. I am going to resort to the original thought experiment by Einstein, involving the moving railway carriage and two lightning strikes. Now since the observer (A) in the railway carriage moves towards one lightning bolt and away from other, he shouldn't perceive the strikes being simultaneous. Whereas for an observer (B) who is completely at rest on a platform the strikes are simultaneous.
However this description is valid in the reference frame of observer (B). My question now is that - Is there any symmetry in simultaneity between the two observers (just like time dilation and length contraction)? For all we know, in the reference frame of railway carriage, observer (A) is at rest and observer (B) is moving in the opposite direction. Hence, the two events are simultaneous for (A) in his reference frame. Does this imply both observers perceive the events to be simultaneous in their own reference frames?
 A: There is a symmetry in the following sense :

*

*two lightning strikes which appear simultaneous to $(B)$ appear non-simultaneous to $(A)$


*two lightning strikes which appear simultaneous to $(A)$ appear non-simultaneous to $(B)$
However, for one particular set of events (a series of two lightning strikes), they can only appear simultaneous to one observer.
A: I understand your doubts; they come from our ingrained Newtonian way of thinking. There could also be a situation in which observer $A$lice perceives the two lightning-strike events as simultaneous; but in that case they would be perceived at different times by $B$ob.
It may be good to emphasize that, in this specific case, both $A$ and $B$ agree that two light signals, starting from the lightning-bolt events, reach $B$ at the same time: that is, the event "light signal from bolt 1 reaches $B$'s eyes" and the event "light signal from bolt 2 reaches $B$'s eyes" are coincident events. Also, both $A$ and $B$ agree that two other light signals, starting from the lightning-bolt events, do not reach $A$ at the same time: that is, the event "light signal from bolt 1 reaches $A$'s eyes" and the event "light signal from bolt 2 reaches $A$'s eyes" are not coincident events. These are two facts experimentally verifiable by both $A$ and $B$.
The question is how $A$ and $B$ interpret these two facts upon which they agree.
According to $A$, the two light signals don't reach her at the same time because they were not generated at the same time, since both travelled with speed $c$ for equal distances. And they reach $B$ at the same time because $B$ is moving towards the signal that originated later – and so that signal travelled for a shorter distance. Note that $A$ deduces that the two signals reaching her cannot have originated at the same time. This is a consequence of the fact that all observations are in the end local observations at the observer's place, combined with the postulate or convention that light has the same speed for every observer.
According to $B$, the two light signals don't reach $A$ at the same time because they were generated at the same time, and both travelled with speed $c$, but $A$ was moving towards one of them – and so that signal travelled for a shorter distance. And they reach $B$ at the same time because he is not moving. Note, also in this case, that $B$ deduces that the two signals reaching him must have originated at the same time.
I hope that this description shows you the symmetries and "antisymmetries" in $A$'s and $B$'s interpretations. Check out also the other case, in which the two bolts are simultaneously seen by $A$, and therefore not by $B$.

An important point to be emphasized at the start is that according to $A$  the two lightning-strike events happened at equal distance from her. This is the case because, for example, whenever $A$ sends two light signals towards the two ends of the rail carriage when her clock shows some time $t'$, both bounced signals reach her at the same time $t''>t'$ according to her clock (this is the notion of radar distance).
But the same is true for $B$: the two lightning-strike events happened at equal distance from him. Because whenever he sends two light signals towards the two points on the rail track – where the lightning bolts landed – when his clock shows some time $t'$, both bounced signals reach him at the same time $t''>t'$ according to his clock.

Personally I find it easier to avoid thinking in Newtonian terms by considering the following experimental fact and its consequences: You, I, and even a third person are at the same spot and synchronize our clocks; we then move around freely in different ways. If we happen to meet again in some spot and compare our clocks, we will find that they generally give three different readings. How can we decide which clock "is right"? We can't. This forces us to redefine our notions of time and space, and makes it impossible to associate "time" to places or events – it can only be associated to observers.

For the notion of radar distance see for example

*

*Frankel: Gravitational Curvature: An Introduction to Einstein's Theory, chapter 2;


*Landau, Lifshitz: The Classical Theory of Fields (also here), § 84.
A: To supplement the answer by @SolubleFish ,
it might be helpful to interpret "simultaneity" geometrically,
in terms of the tangent line to a circle ("the curve of constant square-interval").
In Euclidean geometry,
a "surveyor" traces out a radial line.
Where the surveyor meets the circle of radius t (read off an odometer),
consider the tangent line there.
The surveyor will assign the same odometer value of t for all points on that tangent line.
Note that according to the red surveyor, the red point and the black point are on the same tangent line, but not the green point.

Similarly, according to the green surveyor, the green point and the black point are on the same tangent line, but not the red point.

(This could be called the relativity of being co-tangential).

(For an arbitrary black point, the circle-radii differ among surveyors.
For an arbitrary radial surveyor through the origin, his family of tangent lines are parallel.
But the important conclusion remains true.
Two distinct points on one surveyor's tangent
are not [generally] on another surveyor's tangent. )
For Minkowski spacetime, the construction is analogous.
In this case, the "circle" is a hyperbola.
The inertial observer traces out a radial line.
"Space at time t" for that observer is defined by the tangent line [hyperplane]... since that observer will assign the same value of "t" for all events on that tangent line.
So, analogously (in accord with @SolubleFish ),
according to the red observer, the red event and the black event are simultaneous, but the green event is not mutually simultaneous with them.
Similarly, according to the green observer, the green event and the black event are simultaneous, but the red event is not mutually simultaneous with them.

(This is the relativity of simultaneity).

In the Galilean case (which is an exceptional case in this viewpoint),
since the circle is a line [hyperplane],
the family of tangent lines of the red observer coincide with that of the green observer

(This is absolute simultaneity).

These diagrams are based on my
https://www.desmos.com/calculator/kv8szi3ic8 (robphy's spacetime diagrammer for relativity v.8d-2020 ). Tune the E-slider to see the various cases (E= -1 for Euclidean; E= +1 for Minkowski; E= 0 for Galilean).
A: To elaborate on pglpm's answer, the real issue is that "simultaneity" isn't really an objective observer-independent physical concept in special relativity.  Instead, the objective relativistic concept that most closely corresponds to simultaneity in Newtonian physics is spacelike separation.
Specifically, in Newtonian physics, two events (i.e. points in spacetime) A and B can be ordered in three different ways, which all observers will agree on:

*

*A may happen before B;

*A may happen after B; or

*A and B may happen at the exact same time.

In special relativity, as you've learned, this is no longer true.  While we can still define such an ordering of events in the reference frame of any particular observer, in special relativity, as you've learned, observers moving at different velocities can disagree on the relative order of events.
However, we can instead set up an alternative trichotomy that all observers can agree on even in special relativity:

*

*A may be in the past light cone of B, i.e. light* emitted by A can reach event B;

*A may be in the future light cone of B, i.e. light emitted by B can reach event A; or

*A and B may be spacelike separated, meaning that they occur sufficiently close together in time and sufficiently far apart in space that light from neither event can reach the other.

*) By "light", I actually mean any causal influence propagating at $c$ or slower. Also, in asserting that the three options above are mutually exclusive, I'm implicitly assuming that A and B are not the same event, i.e. that they do not occur at the exact same time in the same place.
Spacelike separation can be seen as a generalization of simultaneity, in the sense that distinct simultaneous events (according to any observer) must always be spacelike separated, and also in the sense that for any two spacelike separated events there is a reference frame from which those events appear simultaneous.
Notably, all observers in special relativity, regardless of their velocity, will always agree on whether two events are spacelike separated or not.  (And, if not, they'll also agree on which of the events occurs first, in the sense of being in the past light cone of the other.)
Also, since special relativity forbids influences propagating faster than light, it follows that spacelike separated events are causally independent.  Thus, the choice of which one of two spacelike separated events we wish to regard as happening "first" is indeed an arbitrary one with no physical significance: no information can travel between such events in either direction and neither one can exert any physical influence on the other.
