What physical construction features in a transformer control the dot polarity?
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.