Average two signed integers without overflow

The safe-midpoint trick lo + (hi - lo) // 2 assumes lo <= hi and non-negative indices. But suppose you need the floored average $\lfloor (a + b) / 2 \rfloor$ of two arbitrary signed integers, either of which may be near the 32- or 64-bit limit, and you cannot assume any ordering or sign.

Compute $\lfloor (a + b)/2 \rfloor$ with no intermediate that could overflow a fixed-width integer.