This solution occurred to me:
For Each cell In Selection    'Determine current reference type
    Select Case cell.Formula
        'xlAbsolute
        Case Application.ConvertFormula(cell.Formula, xlA1, xlA1, 1): ref = 2
        'xlAbsRowRelColumn
        Case Application.ConvertFormula(cell.Formula, xlA1, xlA1, 2): ref = 3
        'xlRelRowAbsColumn
        Case Application.ConvertFormula(cell.Formula, xlA1, xlA1, 3): ref = 4
        'xlRelative
        Case Application.ConvertFormula(cell.Formula, xlA1, xlA1, 4): ref = 1
    End Select
    
    'Cycle reference type
    cell.Formula = Application.ConvertFormula(cell.Formula, xlA1, xlA1, ref)
Next