You generally want complementary NPN/PNP pairs when they are used as opposites of each other in the same circuit. One transistor is usually used with opposite polarity as a mirror image of the other.
For example, consider this circuit:
Q1 drives the output during the positive half of the waveform and Q2 during the negative. Clearly the requirements on each transistor are the same. If you were to put a sine wave in, you'd want the top and bottom halves of the output to be mirror images of each other. This means that not only will the two transistors be subjected to the same voltages and currents, but it is also useful for them to have the same gain and other characteristics.
Manufacturers know this, so often explicitly say that a particular pair of models are "complementary" of each other. Not only does that save digging thru datasheets looking for two that have close to the same specs, but you also know that the manufacturer deliberately intends them to have the same specs.
Of course the flip side of this is that well-designed circuits are reasonably tolerant of variations in parameters like gain. Note that in the above circuit, R1 and R2 reduce how much the overall transfer function depends on gain. Still there will be some dependency, and having both halves of the circuit respond equal and opposite of each other remains useful.
If you can cite other configurations that need complementary pairs, I'm interested to know
As I said above, you generally want complementary pairs when each transistor is half the circuit, and they are operating as mirror images of each other.