As we know, the dot convention is used on transformer symbols to denote whether the transformer "inverts" the signal passing through it or not. However, I am not clear on what in the physical construction of a transformer determines the dot polarity -- is there something physically different between two otherwise identical transformers that have opposite dot polarity conventions? (I.e. one has the dots on the same side and the other has the dots on the opposite side) Or are these two transformers physically identical and interchangeable?
The primary of a transformer and the transformer core are a electromagnet. If you put current thru the primary one way, the core is magnetized one way. If you reverse the current, the magnetization direction is reversed. Likewise, the secondary produces a voltage based on how this magnetism changes.
Between how the primary and secondary are connected to their circuits, there are two opportunities to flip overall polarity. You can flip the primary connections, or the secondary connections. Each causes a flip, so flipping both gets you back to where you were.
The winding direction of a coil determines the magnetism direction for a fixed current direction. Just like flipping the two leads of a winding, you can get the same effect by unwrapping the winding, then wrapping it back with the opposite twist.
To actually answer your question, the dots indicate relative polarity between the primary and secondary. This is fixed for a particular transformer by the direction each coil is wound, and by how the leads are labeled.
Winding direction determines the output polarity.
Figure 1. Double-primaries and double secondaries on transformer.
The dots tell you that the primaries and secondaries are wound the same direction starting from that end of the coil. When connected in series their voltages will add.