When can I react during my opponent's turn, and can a player play a card from hidden (like Consult the Past) and immediately reveal it to use its effect on the same turn?

Ruling: You cannot play a hidden card on the same turn that you hide it. When playing cards from hidden on a later turn, they function exactly like playing from hand - spells and abilities go on the chain and grant reaction windows, but permanents (units/gear) resolve immediately without a reaction window.

Sequence:

• Hide a card on one turn
• On a subsequent turn (not the same turn), you may play it from hidden
• If it's a spell or ability, it goes on the chain and opponents can react
• If it's a permanent, it resolves immediately with no reaction window (unless it has a triggered ability)

Nuances:

• Playing any card (from hand or hidden) opens a chain, but permanents leave the chain immediately during the Finalize step, giving no opportunity for reactions
• You can only react to spells, triggered abilities, and activated abilities (with small exceptions for 'add' abilities)
• Permanents may have "When You Play Me" (WYPM) triggers that do grant reaction windows, but you're reacting to the triggered ability, not the permanent itself
• During showdowns, there are additional focus/priority rules that allow playing cards