Coil Winding Sense of Inductor/Transformers and Output Polarity

Question

Given a solenoidal coil, if it is wound in one sense (say clockwise), would the output from it, in terms of polarity, be equivalent to another coil, wound in the opposite sense (counter-clockwise), with its lead also swapped? In other words, does switching the leads from an inductive coil produce the same effect as changing the winding direction of the the coil?

For an inductor, this primary instance where this would come into play is if the coil were used as an antenna of an inductive RF signal; would the polarity of the RF signal and the polarity of the voltage coming from it match or not in accordance with the right hand rule?

For a transformer, this is slightly more complex, because now it is the relative winding sense of one coil with respect to the other, the primary to the secondary. For instance, if both coils were wound clockwise or counterclockwise, would the polarity from one coil to the next would correlate in one way, but if one coil was wound clockwise and the other counter-clockwise, would the polarities correlate in the opposite way?

Here is a diagram to illustrate this question:

A mathematical (or at least graphical) proof of this would be helpful.

Answer 1

Changing the current flow direction in the windings of a coil is the same as reversing the winding rotation direction.

For a transformer, if both coils are wound identically then the polarity in the secondary will match the polarity in the primary. Imagine you wind two parallel windings and connected the two coils together - you wouldn't suddenly find that there is any problem with this - litz wire relies on this of course. If you open circuited one of those paralleled coils while still applying voltage to the other, you'd find the voltages were the same polarity and virtually indisinguishable for each other when tightly coupled.