Why does the vertical resistor have no voltage across it? I'm trying to calculate the resistance between A and B where each resistor between them has resistance R. According to my key, the middle, vertical, resistor has no voltage across it, and thus is negligible in the calculation. Why is this the case? How can we "see" that it will have 0 voltage?

It is stated that symmetry implies identical currents. But why would the following circuit be impossible?

 A: You can see this by realizing the upper and lower path are completely symmetric. Thus the current through each of them must be the same. And if the same current is flowing through resistors with the same resistance (the two diagonal ones on the left), the voltage drop across them is exactly the same. Therefore, the topmost and bottom-most corner are at the same potential, and the vertical resistor is connecting points of equal potential, resulting in no current across it.

In general, it's good advice to take advantage of symmetries in a problem. If a problem is symmetric under some operation (in this case, flipping the diagram upside down), the solution very often must also be symmetric under that same transformation. Be careful though, there are also cases where symmetry spontaneously breaks (phase transitions for example), but looking for symmetries is good practice.
A: In your bottom diagram, let's say that the voltage at $B$ is zero, for convenience.
Let's label the top point as $x$, and the bottom point as $y$.
Ohm's law states that the voltage at $x$ must be
$$\frac{IR}{6}$$
And the voltage at $y$ must be
$$\frac{IR}{2}$$
But then this would mean that the current through the middle resistor must be
$$\frac{\frac{IR}{6}-\frac{IR}{2}}{R} = -\frac{I}{3}$$
(Negative meaning it would flow from $y$ to $x$).
Which it can't be, because our beginning assumption is that it was
$$\frac{I}{6}$$
