As performing jazz pianist I am with Renato and Dave very pessimistic
about the classical music training here in The Netherlands. In the 50ies
and sixties classical musical students who also played jazz on another
instrument (I worked with a now reasonably known classical pianist, who
played jazz on the trombone) were threatened to be removed from the
conservatory! I have many classical pianist friends who are very jealous
as I play anything that comes into my head and, just as david mentioned
sometimes improvise on short themes they provide and then hear them
struggling to do the same.
However, there are profound differences in technique and key-touch
between classical and jazz pianists (the same goes for other
instruments). I have written an arrangement for solo piano of Duke
Ellington's Sophisticated Lady and given to some of them to perform.
It is however rather difficult for them as (A) the chords are outside
their normal repertoire and (B) rhythmic interpretation of the text is
typically "non-jazzy", something you can hear from the first note.
Julian, you know I presume that 2 very great jazz musicians were blind
(Art Tatum and Lenny Tristano) you should listen to them and be amazed
about the techniques they use.