What happens with the tip vortices of two aircraft flying in the opposite direction? Suppose that an aircraft flies with its wing through an tip vortex of another aircraft which flew in the opposite direction. Suppose that the shed wing tip vortex of both aircraft are exactly the same in strength, only rotating in opposite direction?
What will the net effect be? Will the angular momentum be canceled?
I would like to place this in the light of the existence of a root vortex experienced by wind turbines. From Burton's Wind Energy Handbook (3rd edition, p78) :
"
For example, on a two blade rotor, unlike an aircraft wing, the bound circulations on
the two blades shown in Figure 3.27 are opposite in sign and so combine in the idealised
case of the blade root being at the rotational axis to shed a straight line vortex along the axis with strength equal to the blade circulation times the number of blades.
"

With the aircraft example the circulation is opposite in sign as well. So why is the root vortex a summation of the two blades instead of cancellation?
 A: 
Suppose that an aircraft flies with its wing through an tip vortex of another aircraft which flew in the opposite direction. Suppose that the shed wing tip vortex of both aircraft are exactly the same in strength, only rotating in opposite direction?


What will the net effect be? Will the angular momentum be canceled?

Vortices are described as vector fields. If you superimpose two vector fields on top of each other, vectors in the same location of equal magnitude but opposite direciton will sum to zero.
Therefore two perfectly overlapping and opposite but otherwise identical vortices will sum to zero everywhere and perfectly cancel each other out. However, in your scenario the two airplanes would clip wings or collide causing the vortices to stop short of overlapping each other. And even if the airplanes flew through each other the airplane itself would disrupt the pre-existing vortice from the first plane. And even if the airplane somehow did not the vortice produced by the plane passing by would be stronger than the vortice left behind by the other plane since it dissipates.
But if you had two air vortex guns pointed at each other and shooting, they would perfectly cancel only out at the midpont between the two vortex guns since the vortices would have had the same amount of time to dissipate and therefore be of equal strength.

With the aircraft example the circulation is opposite in sign as well.

I think this is horribly written. Let's first establish that that the vortices between the two airplane wing tips of an airplane are in opposite directions. This should be obvious.
On a two-bladed propeller I would consider the vortice generated by each blade to be in the same direction relative to the blade that is generating it (this is not true of the airplane). So the author's wording really throws me for a loop even with the diagram.
The author is talking about is talking about the swirl direction of the two vortices at the same location along the central axis of travel. So basically the direction of the two vortices generated by blades at the same point in time.
In this case they aren't perfectly overlapping but they do overlap. It's a complex path of a swirling vortex that itself has a helical path.

So why is the root vortex a summation of the two blades instead of cancellation?

"Summation" and "cancellation" can have different meanings. I don't exactly what you mean when you say "cancellation" here, but many would interpret cancellation as a complete cancellation. That can't happen here since the vortices are not perfectly overlapping and opposite but otherwise identical. The result cannot be zero.
"Summation" tends to be much more general. You can have a summation of two vector fields and have them all cancel to zero. That is still a summation. But more often you have a summation where certain parts cancel and other parts do not cancel. That is what is happening here. Each blade produces a vortice that has a helical, corkscrew path and the vortice itself swirls around that helical path. If you sum of the vector fields of both tip vortices the net field you get is one known as the swirl of the prop wash.
A: I'am not sure of the wisdon of flying directly at another plane, but since the vortics on the left and right wingtip are opposite to each other, a plane flying with its right wingtip just behind the left wingtip of a plane immdeiately ahead will zero out  the vortex and reduce the KE in its trailing vortex and so reduce its own drag --- a fact well know to geese   who fly in V formations for this reason.
A: Now let's assume the two airplanes do not collide but two of their trailing vortices happen to have their core along the same line. Yes, those two vortices will cancel each other.
But you forget that both planes have two trailing vortices, one on the left and one on the right side. So what is left after half of those vortices  cancel each other are two vortices at twice the distance of the vortices of one airplane.
If we neglect dissipation and assume that those two vortices will stay there for some time, they will look just like the vortices left by a single airplane of twice the wingspan and four times the mass of those two which did not collide. However, the downwash between the remaining vortices will look differently and reveal that two separate airplanes were involved in its creation.
