Show Two Hands, Play One

In a showdown variant of rock–paper–scissors, each player raises both hands, showing one throw on each, so all four throws are visible. Then both players simultaneously withdraw one hand, and the two remaining hands decide the round for $\$1$ (win $+1$, loss $-1$, tie $0$). You are showing Rock and Paper; your opponent shows Rock and Scissors. What is optimal play for each side, and how much is the round worth to you?