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Say I, on my opponent's turn, have Infiltrate facedown and they play a secret action that has an orange icon (I, of course, don't know this). How does this interaction work, considering secret actions are considered actions?

Infiltrate will never affect a Secret Action because as youve said, you cant see its Battle Icons as its played.
Source: Wizards of the Coast (2019-05-28)    【Link to Ruling】
Tags: Battle Icons