FX options as calls and puts at once

Why can a foreign-exchange option be described either as a call or as a put? How does this observation relate to the result about early exercise of American calls?

A call on euros paid in dollars is the same contract as a put on dollars paid in euros. The right to buy euros at a fixed dollar price is the right to sell dollars at a fixed euro price. A foreign currency is like an asset paying a dividend (its interest rate), so the no-early-exercise result for calls on non-dividend stocks does not apply, consistent with the fact that the equivalent put can be optimally exercised early.