Gene Hackman -- Twisted Wreckage from Bloody Bike Crash

You're looking at the mangled remains of the bicycle Gene Hackman was riding when he was struck by a car last week ... remains that Hackman has decided to KEEP as a memento.

TMZ has learned ... Hackman's wrecked Trek mountain bike was brought to the Key Largo Bike Tours and Adventures shop days after the January 13 accident in the Florida Keys.

Sources tell us ... judging from the condition of the bike, it appears Hackman was hit from behind, because the back tire was almost completely folded in half from the impact.

We're told Hackman's people asked the shop to repair the bike ... and replace all of the damaged parts ... but they made it clear, Gene wants to KEEP all of the broken pieces ... including the twisted tire.