You must always allow other players to examine a card when some aspect of the face of the card is being referenced, such as when a specific card is taken from your draw deck.
You may inspect your own cycle pile and eliminated pile, which is helpful in determining if a particular card is in your draw deck. You may always inspect face up cards in play.
You may not show any other players your hand. Conversations between teammates must take place at the table, in public. You may talk to your teammate about what cards you have, but that means the other team will also know.