Wiring a transformer's coils together?
This wouldn't work very well, but it would work.
A transformer is designed to keep as much flux as possible contained inside its core (microwave transformers specifically are an exception to this, so it might work better than most, but it's still optimized for use as a transformer, not an inductor.), while large-value inductors rely on having some sort of air gap to avoid saturation and increase energy density; most of the energy in an inductor is stored in the air gap.
As for winding direction, you can always swap the wires to correct a mismatch. If they're wound oppositely and connected in parallel, the flux would just cancel and you'd get a lower inductance than you would otherwise, but like I said, you can just swap the wires.
As an additional point, to maximize inductance you should connect the windings in series, not in parallel. Connecting them in parallel would lower the inductance compared to putting them in series.