After doing some reading, it seems we can make the following observations which answer the question:
- There are differences associated with the channel at either end of the MOSFET.
Strain: This technique involves altering the crystal lattice structure of the silicon in the channel region to improve carrier mobility. The application of strain can be different near the source and drain regions, leading to asymmetrical electrical characteristics. For instance, tensile strain might be applied near the drain to enhance electron mobility in nMOS devices, which can make the drain region's electrical properties slightly different from the source. Incidentally, there are various reasons you might introduce more strain at the drain end than at the source end, but whatever these reasons they lead to a difference between the source and drain.
Halo Doping: Halo or pocket doping involves implanting impurities in the substrate around the source and drain regions to control short channel effects like threshold voltage roll-off and punch-through. The concentration and distribution of these dopants can differ between the source and drain, leading to asymmetries. For example, heavier doping near the drain side can be used to manage hot carrier effects, which introduces a difference between the source and drain.
- There is another category of differences associated with geometry and layout of the diffusions.
Physical Dimensions: The actual physical dimensions of the source and drain regions can be intentionally varied. For example, in some power MOSFET designs, the drain region is made larger to handle the higher voltage drop, while the source is optimized for better on-state resistance.
Parasitic Elements: The layout of the MOSFET on the chip can lead to different parasitic capacitances and resistances at the source and drain. For example, if the drain is located farther from the gate than the source due to layout constraints, this can lead to higher drain-gate capacitance. (This point was referenced in my question but it is not really intrinsic to device operation. That is, if we "flipped" the device around in an otherwise symmetrical textbook MOSFET we would get the same operation.)