If you have two identical but oppositely wound inductors does the inductance effectively cancel?

It depends of the extent of their mutual inductance. Roughly speaking, if they are wound on the same ferrite core, they have a large proportion of their inductance that is mutual (as in a transformer for instance), so they would mostly cancel. If, as it seems to be in your case, they are separate components, they are wound on separate cores and have little mutual inductance. In case of air inductors or resistors as mentioned, the distance between them and exact geometry has to be taken into account and only an electromagnetic simulation software can tell how much is canceled, but that would be much less than a complete cancellation.

If you put two inductors in series with a common axis, there will be little cancellation. If the two were put into close proximity side-by-side you'd get more, but probably not much.

It is possible to get non-inductive wire-wound power resistors. These essentially take a single conductor, fold it in half, then wind the resulting doubled wire on a core (not a magnetic core). Because the halves are physically close, the voltage rating of the resulting resistor is less than with normal construction (winding the wire on a core), but the inductance is much less.