Commuters facing off on a narrow platform

A line of commuters stands single-file on a narrow platform. Each person faces either left or right. Whenever a right-facing person stands immediately to the left of a left-facing person, so the two are looking straight at each other, a pattern we can draw as → ←, they get flustered and, in the next moment, both turn around, becoming ← → (now looking away from each other).

Several such face-offs may exist at once, and they may keep triggering new ones as people spin around. It looks like it could churn forever.

Prove that the turning must eventually stop, the line always reaches a state with no one facing anyone else, regardless of the starting arrangement or the order in which face-offs are resolved.