My opponent used a Recycle but I didn't see the card they put on top of their deck. Do they have to show it to me even though it does not say that on the card any longer?
Yes, even when it is not printed on the card, you must still clearly indicate which card you are moving because your Discard Pile is open to all players.
Source: TPCi Rules Team (2011-12-01)

Relevant cards

Note: All applicable cards may not be listed below.
BW2_96
WOTCFO_61