[nfredin]

Is the actual structure of an amine hydrochloride salt just the aminium ion with a chloride counterion (H covalently bound to N and N+ ionically associated with Cl-, as opposed to H covalently bound to Cl)?  If so, why is it so seldom represented that way (e.g., triethylammonium chloride rather than triethylamine HCl)?

[voidSetup]

Yes it is an ammonium ion with a chloride counterion.  The H protonates the nitrogen of the amine, and the positive charge lies on the nitrogen.  Since it is a salt is has an ionic bond (the N-Cl bond). I think sometimes people would call it something like triethylamine HCl to show that it was crystalized using HCl.  You could use something else as the counterion.