By coincidence, I came here to ask the exact same question. Here’s what I’ve figured out:
Conclusion
The rotation tool uses the center of the selection as the pivot, which leads to off-grid movements.
The rotation tool can be forced to use a specific footprint as the pivot by first using the move tool with “Warp mouse to origin of moved object” enabled, and then switching to the rotation tool.
If the pivot is on the desired grid, and the desired grid is active, then the footprints in your move or rotation will also be on the desired grid.
If you are not careful to meet both these conditions, you will have off-grid footprints.
Experiments
Setup
First off, you’ll want to make sure that “Warp mouse to origin of moved object” is set (which it sounds like you do):
Anyway, starting out with a few resistors on a 1mm grid:
Rotated, they’re still on the grid:
Let’s offset one of them 0.5mm:
Rotate around group center
Select them all, press “R”, 
In this case, the rotation happens around the centerpoint of all the footprints combined, and ends up way off-grid.
Rotate around footprint center
Ok, let’s revert, select all, hover over the center footprint, press “M”, press “R”, 
In this case, pressing M snaps the cursor to the nearest footprint’s centerpoint, the middle footprint. The rotation then happens around the cursor. The middle footprint is on the 1mm grid.
Let’s try that again, but instead let’s make the nearest footprint be the .5 footprint,
In this case, the rotation again happens around the cursor, but since the closest footprint to the cursor is the footprint on the 0.5mm grid, the other footprintss are now off-grid.
Translate around footprint center
From a reset state, hover over the middle footprint, press M, and move
From a reset state, hover over the right footprint, press M, and move
