That seems to me to imply that in the game, the reverse card turns things around on the other player. Like if they play a +2, you can play the reverse card and make them pick up 2. Is that the way some people play?

Exactly, as I am used to, action cards do add up: E.g. if someone (player A) plays “draw 2”, the next player (B) can lay another “draw 2” on top without drawing 2 cards themself forcing the next player © in line to draw 4 if they themself don’t lay another “draw 2” or a reverse card of matching colour. The latter would then chang the direction of play and force player B to act accordingly.