Space (Lorentz-Fitzgerald) contraction Lorentz transform can be written, according to wikipedia, as:
$$
\begin{align}
  ct' &= \gamma \left( c t - \beta x \right)  \\ 
  x' &= \gamma \left( x  - \beta ct \right)  \\
  y' &= y \\ 
  z' &= z.
\end{align}
$$
I see a complete symmetry between terms ct/ct' and x/x', that is, I can exchange ct with x and ct' with x' obtaining exactly the same equations. 
However, books talk about "time dilation" and "space (Lorentz-Fitzgerald) contraction", not about "time and space dilation".
I assume "time" is related to term ct and "space" to x.
Please, could someone help me to see my error or understand the point of view under the expression "space contraction"?
 A: Although the treatment of $x$ and $t$ in the Lorentz transformation is exactly analogous, the symmetry is broken by the way we measure distance and time. To measure proper time, we allow a clock to exist along some timelike world-line. To measure proper length, we have to carry out a much more complicated process, including verifying that the the measurements at the ends of the measuring rod are simultaneous. To verify this, we need to send signals back and forth.
The clock's world-line is one-dimensional, and an observer can be with the clock at all times. The measuring rod's world-line is two-dimensional, and the observer can only be at one part of the rod at one time.
A: To measure the length of a rod in the frame of reference in which it is stationary (the S' frame or "the rod's frame") we place a ruler against it and take readings against the ruler of either end of the rod. It doesn't matter whether or not we take the readings at the same time.
To measure the length of the rod in the laboratory or S frame, in which the rod is whizzing past the ruler we must clearly take simultaneous readings of either end of the rod against the ruler. Thus the events of taking the readings are at (say) ($T, x_1) \text{and} (T, x_2)$.
The separation of these events in the ruler's frame (where the events are not simultaneous, but this doesn't matter) is$$x_{1}'-x_{2}'=\gamma(x_1-\beta cT-[x_2 -\beta cT])= \gamma(x_1-x_2 )$$ 
So $x_1-x_2 < x_{1}'-x_{2}'.$ Hence the length contraction.
Now consider time dilation. A rocket whizzes through the laboratory (the S frame). Let it emit flashes of light separated by a time interval $T_0\  (=t_{2}'-t_{1}')$ as measured by its own clock. In the clock's frame (the frame in which the clock is stationary) these flashes are in the same place (X'). In the lab frame they are in different places. Using the inverse transform, the time interval between the flashes, as measured in the lab by (synchronised) clocks at the places where the flashes occur, is
$$ct_2-ct_1=\gamma(ct_{2}'+\beta X'-[ct_{1}'+\beta X'])=\gamma(ct_{2}'-ct_{1}').$$
Thus $t_2-t_1\ >\ t_{2}'-t_{1}'.$  This is time dilation.
We can now see how the asymmetry arises. For length comparisons, measurements must be at the same time in the lab frame. For time comparisons the place is the same in the 'moving' frame, but different in the lab frame.
A: Even you can derive those 2 equations in terms of each other, they mean different things. Its called time dilation and space contraction because they mean seperate things. Space and time does mot respond to same at relativistic speeds. While lenghts gets shorter (contracts) time passes more slower (dilates)
