I've got a feeling this will be a very unpopular proposal to both fanbases, but in terms of team needs, it does make sense.
The Twins have fallen out of contention and could look to rebuild. Mauer would obviously net a large return in terms of high-level prospects. Sticking with the one-for-one format, though, he would be deserving of one of the Giants' top pitchers in an equal swap.
The Twins would probably start by asking for Tim Lincecum, but probably would not get far in those discussions.
Matt Cain could be a more realistic target, though. Cain is every bit as important to the Giants as Lincecum, but he lacks the gate-appeal and star power that Lincecum possesses.
Cain is as reliable and dominant of a pitcher as there is the majors and he was downright unhittable last postseason as the Giants advanced to become world champions.
The Giants have depth in the rotation, though, with Jonathan Sanchez and Madison Bumgarner behind Lincecum. Barry Zito should return from the disabled list soon and Ryan Vogelsong has pitched well as Zito's replacement. Losing Cain would sting, but the Giants could absorb the loss with the added offense that Mauer would provide.
With the season-ending injury to backstop Buster Posey, the Giants need a catcher. Joe Mauer has had his own season shortened due to injuries, but will be back and likely productive much sooner than Posey.
If the Giants attempt to convince their young star to switch positions, they will need a long-term solution behind the plate. There are few, if any, better options behind the dish than Mauer.
Mauer at catcher, Posey at first and Brandon Belt in the outfield solves the Giants' lineup problems for a long time.
Mauer is expensive, at $23 million per year through 2018, but the Giants can afford to take on his contract.
Meanwhile, if the Twins do opt to rebuild, Cain is much more affordable at $7 million in 2011 and $15 million in 2012.
If the Twins were able to extend Cain beyond 2012, he could be the ace the Twins build their rotation around to compete in the AL Central.
Would Minnesota take a one-for-one swap for Mauer? Even if it were for a pitcher of Cain's caliber, I am not sure. On paper, at least, it makes sense, though.