Thank you, wilson, these work nicely!

It also works for columns; the following sub combines the functionality for all four directions:

Sub MoveRowsOrColumns(direction As String)
    Dim rOriginalSelection As Range
    
    Select Case direction
    Case "up", "down"
        Set rOriginalSelection = Selection.EntireRow
    Case "left", "right"
        Set rOriginalSelection = Selection.EntireColumn
    Case Else
        Debug.Assert False
    End Select
    
    With rOriginalSelection
        .Select
        .Cut
        Select Case direction
        Case "up"
            .Offset(-1, 0).Select
        Case "down"
            .Offset(rOriginalSelection.Rows.Count + 1, 0).Select
        Case "left"
            .Offset(0, -1).Select
        Case "right"
            .Offset(0, rOriginalSelection.Columns.Count + 1).Select
        End Select
    End With
    Selection.Insert
    rOriginalSelection.Select
End Sub

Here is a sub that also works for columns; it combines the functionality for all four directions:

Sub MoveRowsOrColumns(direction As String)
    Dim rOriginalSelection As Range
    
    Select Case direction
    Case "up", "down"
        Set rOriginalSelection = Selection.EntireRow
    Case "left", "right"
        Set rOriginalSelection = Selection.EntireColumn
    Case Else
        Debug.Assert False
    End Select
    
    With rOriginalSelection
        .Select
        .Cut
        Select Case direction
        Case "up"
            .Offset(-1, 0).Select
        Case "down"
            .Offset(rOriginalSelection.Rows.Count + 1, 0).Select
        Case "left"
            .Offset(0, -1).Select
        Case "right"
            .Offset(0, rOriginalSelection.Columns.Count + 1).Select
        End Select
    End With
    Selection.Insert
    rOriginalSelection.Select
End Sub