Why doesn't an inductor mind polarities?
There are inductors with polarity. They have a permanent magnet to bias the inductor opposite to the expected DC current. It's generally not worth it, but it has been done. 

See, for example, DOI: 10.1109/ESTS.2011.5770892 Permanent magnet inductor design. 


  Abstract:
  Permanent magnet inductors (PMI) are useful in dc biased applications. They utilize the flux produced by a permanent magnet (PM) to partially offset the flux established by current in the inductor winding. The addition of permanent magnets allows for decreased inductor mass for a given inductor current and inductance rating. Herein, the Pareto-optimal front between loss and mass is established for a PMI inductor and is compared to that of a traditional EI core inductor. It is shown that the PMI inductor is significantly less massive than the EI core inductor for a given loss.


Polarized capacitors (which is a significant limitation on their application) are used because a polarized capacitor can be made significantly smaller and lighter than a non-polarized capacitor of the same  value. If non-polarized capacitors were the same size, cost and performance as polarized capacitors of the same capacitance and voltage rating, there would be no reason to use the latter.