Comments/Ratings for a Single Item

The obvious possibility for the number that would be squared to value a
doublemove piece would be the piece's mobility; or rather, you'd expect
the mobility of a doublemove piece to be related to the square of the
regular piece's mobility (but lower, due to overlap).

The average mobility of a Knight on an 8x8 board is 5.25.  (This counts
moves to occupied spaces, on the grounds that the move either threatens or
protects a piece, but naturally not moves off the board.)

Assuming a board where 15% of spaces are occupied by friendly pieces and
15% by enemy pieces (leaving magic number = 70% of spaces empty), a
Double-Knight's mobility is increased by 15.5 for the spaces two N moves
away (I have laboriously calculated), plus 8 * .15 = 1.2 average
'shooting' moves, for a total of about 22, just over four times the
normal Knight.

(That goes up if you count different routes to the same destination as
different moves when you capture different pieces along the way.)

That's not far off from your estimate that a Double-Knight is worth '3 or
4 normal Knights'.  That suggests mobility may be a reasonable predictor
of doublemove value (though it's a royal pain to calculate).

However, the ability to continue after a capture does seem like it should
add quite a bit more than the ability to move through enemy pieces, even
though they result in the same mobility.  So there's probably something
more complex than just mobility changes going on.