In general, "or" is somewhat ambiguous between whether it does or does not include both. In this particular construction---"You may do X or Y"---the tendency is to mean the exclusive or---one or the other, but not both. (But this isn't entirely clear; as these are supposed to be rules for a game, it would probably help to be clearer.)
The rule is unambiguously permissive: it says you may splay them, but doesn't force you do. (If it intended to say that you had to splay exactly one color, a typical phrasing would have been "You must splay either your green or blue cards left.")
It appears you're discussing the game Innovation. (It would, by the way, have been helpful to include that information in the first place.) In that context, I don't know why you think there's any conflict between the two cards. There are lots of cards that cause you to splay different colors; the card you mentioned splays one color, and other cards may splay other colors, and then a later card counts how many colors you've splayed.
From what I remember of the game, I believe in this case the rule is exclusive---color or the other. But you'd be better off checking on forums associated to board games, or that specific game.