Spent a rainy weekend periodically fiddling with this puzzle, trying to find a break in.. I marked the 2x2 region divisions, found and marked with various colors the 4- and 3-chains, but eventually put it on hold until the end of the contest, figuring I'd find the hint I needed from the thread comments. I think I would have eventually realized what each 2x2 quartet belonging to a single leg of the number map implies for categorizing and placing the quartets, but my thought process never really came near the direction required to realize the parity break-in.

However, getting stumped was not discouraging at all, I knew from looking at the puzzle (and from Ivan's past puzzles) that it would eventually be enjoyable, and I patiently waited out the contest duration. Sure enough, by about bullet 3 or 4 of Ivan's great break-in explanation, I was off to the races. And was happy to find that all the cell-coloring work I had done earlier was not wasted, and very useful during the solve process. As expected, an extremely fun solve after getting over that first hurdle. Thank you Ivan!

I believe there is a somewhat simpler way to determine that the L-blocks must all contain only odd numbers.

If one looks at any row (or column) of 2x2 blocks, there are 6 blocks each containing 4 values and 2 L-blocks each containing 3 values (when the 10s are excluded). In these blocks we need to place 10 odds, 8 evens and 12 tens with the restriction that each block can only contain values of the same type. The only way the odds can be accommodated is by using one 4-block and both L-blocks. The remaining groups fit comfortably in 2 and 3 4-blocks respectively.

I still can't understand how anyone can solve this puzzle in 41 minutes! :)