WASHINGTON — A new Republican budget would repeal President Barack Obama’s health care overhaul but put future retirees in a version of Medicare that strangely resembles one of the key cogs in that same plan.

Sound contradictory?

Rep. Paul Ryan, R-Wis., the top GOP budget writer, borrowed the idea of insurance exchanges, a big pooled marketplace, from the health care law enacted in Massachusetts when GOP presidential candidate Mitt Romney was governor. Ryan wants to set one up for Medicare.

Obama borrowed the same idea to make exchanges available to uninsured working families through his law.

Experts who see some agreement rather than irreconcilable differences are scratching their heads and wondering if politicians of both parties may have more in common than they care to admit to voters in the heat of an election year.

“It’s almost as if the roles of the players are being reversed,” said Sen. Ron Wyden, D-Ore. “It is almost ideologically incoherent.” A centrist, Wyden is co-author of a version of the Ryan Medicare plan and also voted for Obama’s health care law. He does not support the spending cuts in Ryan’s latest plan.

“There does seem to be an odd inconsistency on both sides,” said economist Gail Wilensky, who ran Medicare for President George H.W. Bush and advises Republicans.

Ryan’s new budget would begin a gradual shift, reshaping Medicare for retirees, starting with the tail end of the baby-boom generation. The basic government program would remain in place for current beneficiaries and anyone now 55 or older.

But future retirees now 54 or younger would get a payment from the government — called premium support by Republicans and a voucher by Democrats — to pick a health insurance plan through a new Medicare exchange. Private insurance plans would compete directly with a new government option modeled on the traditional program in the Medicare marketplace.

“On the one hand, (Ryan’s) proposal repeals exchanges created under the health reform law to provide coverage for the uninsured, while on the other, sets up new insurance exchanges for already insured seniors,” said Tricia Neuman, top Medicare policy expert for the nonpartisan Kaiser Family Foundation, an information clearinghouse on the health care system.