I've played for a while with Anza Power's simulator and I've managed to swap "1" and "2" in original position.
This means that it is possible to swap any two horizontally adjacent cells, you just have to:
1. Make a couple of obvious moves to bring these two cells in place where "1" and "2" originally are.
2. Apply the procedure that swaps cells "1" and "2".
3. Undo moves made in step 1 (perform "opposite" moves in reversed order).
Exactly the same can be done for every vertically adjacent cells due to the symmetry of the puzzle,
you can easily transform procedure swapping "1" and "2" into procedure swapping "1" and "4".
Now when you know how to swap any two adjacent (horizontally or vertically) cells,
it is easy to solve every possible puzzle: you can just put numbers into their cells one by one.
First put 1 in place, then 2 without touching 1, then 3 without touching 1 and 2, etc.
Alternatively you can solve first three rows using method described by Anza Power and use above for the last row.
All you have to do is to find a sequence of moves that swaps "1" and "2" (possibly by trial and error).
And such a sequence exists, I've checked it.