Inverse direction of copy-dragging formula transposing behavior?

Can this be done?:
Assume C1 is in A1
If dragging A1 horizontally 1 cell over, the formula in B2: C2
If dragging A1 vertically, in A2: D1

Also inverse applying to the $ sign behavior.
C$1 will stay C$1 when dragged horizontally, but change the letter when dragged vertically.
$C1 will change the number when dragged horizontally, but won’t change letter when vertically.

start in A1 with Formula:


Make no sense, because the relative row has no effect on dragging horizontally, and the relative Column has no effect on dragging vertically