Only changing snap to Beat made it clear to me, that odd neither means odd (as in even & odd) nor off the grid (as in not on the grid lines).Select odd (Shift+O) - Selects 'swung' notes. That is notes off the main beat-grid.
I suggest changing the menu entry to something like Select off beat (Shift+O) and the manual entry to Select off beat (Shift+O) - Selects notes not on beat. That is notes 1-and... 2-and... 3-and.