"Actually, I could represent nil as an actual 'cons cell, so that assigning to the car or cdr would work. Crazy?"
That's a bit crazy. :)
PicoLisp does something reminiscent. Every one of its data structures (numbers, symbols, nil and conses) is implemented using the low-level cons cell structure (i.e. a pair of machine words).  They talk about nil's representation fulfilling its dual nature as both a symbol whose value is nil and a list whose car and cdr are nil; both the symbol predicate and the list predicate return true when applied to nil:
: (sym? NIL)
: (lst? NIL)
I'm not sure that they let nil's car and cdr be assignable though, because "NIL is a special symbol which exists exactly once in the whole system."