Chain rings slip when crank is turned
A quick search turned up this post, in which the OP is having the same trouble.

It sounds like your crank is swaged to the chainrings, as in this image:



"Swaged" means the "spider" (the group of radial arms that hold the chainring) is press-fitted onto the crank arm.

A better design is to manufacture the crank arm and spider as one piece, like this:



It seems that on your crank, the connection between crank arm and spider has failed. With the bike upside down and the rear wheel spinning, the load is reduced sufficiently for the crank arm to turn the spider and rings. However, it is just metal-on-metal friction that allows this to happen. As you have seen, if the back wheel is stationary, the crank arm spins independently.

You need a new drive-side crank. I'd replace the whole chainset.

Edit: It sounds like you searched and found the same pages I did, and have gotten confused by the suggestions about Shimano's "freewheeling cranks". Rest assured, your bike will almost certainly not have any weird gimmicky technology like that. Just sounds like a not-so-common failure of a poorly-manufactured crank to me. Swaged cranks are cheap and nasty (no offence intended), but you can replace them with a forged set without breaking the bank.