The only certain plant that is known to have been domesticated in both the Americas and Old World is the sweet potato (Ipomoea batatas). It is native to South or Central America and was first domesticated there in c. 3,000 BCE. The plant is also found from c. 1210-1400 in the Cook Islands, via Polynesian transportation. It's definitely a crop.
The only certain animal species is the wolf (Canis lupus), c. 23,000 BCE in Eurasia, the subspecies Canis lupus familiaris crossing from Siberia to Alaska ~3,000 years later, accompanying humans. This may be considered livestock as dogs were consumed for meat by some American peoples, but I'm not certain if these dogs were Canis lupus familiaris or some other domestic canid.
One could also include the caribou/reindeer (Rangifer tarandus), depending on one's definition of "domesticated". Domestication by Arctic peoples of Eurasian subspecies has occurred multiple times over the millennia. North American subspecies have at least been used by Arctic peoples, though I cannot say if this counts as domestication, semi-domestication, or neither out of concern for not being Eurocentric. Eurasian subspecies were and are kept as livestock, but American ones possibly not.
I believe every other example given here of mutually domesticated species so far is either dubious or a separate species.