Starter problem after having Transmission Rebuild done

and the engine physically tilts over about 1/4 Inch as if the Starter is trying to spin the flywheel, but the Engine does not start.

I would sure like to see this movement of 1/4"!

Who did the R&R of the transmission? Racking my brain here how this could happen and all I can think of by what you are describing is: the engine and transmission are not tightly bolted together. The starter is actually bolted to the transmission case and then engages the engine thru the torque converter bolted to the engine flexplate. I still can't see it though because of the dowel pins locating the engine block to the transmission case so I'm still skeptical here. You might just check how tight the bellhousing to block bolts are.