How to transpose a XML table with T-SQL XQuery?

I suspect there might be a really nice approach using PIVOT, but I don't know it well enough to be able to say for sure. What I offer here works. I have split it up into chunks for better formatting and to provide commentary:

To start with let's capture the example data

-- Sample data
DECLARE @x3 xml

SET @x3 = '
<root>
  <row>
    <column>row 1 col 1</column>
    <column>row 1 col 2</column>
    <column>row 1 col 3</column>
  </row>
  <row>
    <column>row 2 col 1</column>
    <column>row 2 col 2</column>
    <column>row 2 col 3</column>
  </row>
  <row>
    <column>row 3 col 1</column>
    <column>row 3 col 2</column>
    <column>row 3 col 3</column>
  </row>
</root>
'

DECLARE @x xml
SET @x = @x3

-- @x is now our input

Now the actual transposing code:

Establish the size of the matrix:

WITH Size(Size) AS
(
    SELECT CAST(SQRT(COUNT(*)) AS int) 
    FROM @x.nodes('/root/row/column') T(C)
)

Shred the data, use ROW_NUMBER to capture the index (the -1 is to make it zero based), and use modulo and integer divide on the index to work out the new row and column numbers:

,Flattened(NewRow, NewCol, Value) AS
(
    SELECT
        -- i/@size as old_r, i % @size as old_c, 
        i % (SELECT TOP 1 Size FROM Size) AS NewRow, 
        i / (SELECT TOP 1 Size FROM Size) AS NewCol, 
        Value
    FROM (
        SELECT
            (ROW_NUMBER() OVER (ORDER BY C)) - 1 AS i, 
            C.value('.', 'nvarchar(100)') AS Value
        FROM @x.nodes('/root/row/column') T(C)
        ) ShreddedInput
)

With this CTE FlattenedInput available, all we now need to do is get the FOR XML options and query structure right and we're done:

SELECT
    (
        SELECT Value 'column'
        FROM
            Flattened t_inner
        WHERE
            t_inner.NewRow = t_outer.NewRow
        FOR XML PATH(''), TYPE
    ) row
FROM
    Flattened t_outer
GROUP BY NewRow
FOR XML PATH(''), ROOT('root')

Sample output:

<root>
  <row>
    <column>row 1 col 1</column>
    <column>row 2 col 1</column>
    <column>row 3 col 1</column>
  </row>
  <row>
    <column>row 1 col 2</column>
    <column>row 2 col 2</column>
    <column>row 3 col 2</column>
  </row>
  <row>
    <column>row 1 col 3</column>
    <column>row 2 col 3</column>
    <column>row 3 col 3</column>
  </row>
</root>

Works on any size 'square' data. Note the lack of sanity checking / error handling.