  We study a Radon-like transform that takes functions on the Grassmannian of
$j$-dimensional affine planes in $\Bbb R ^n$ to functions on a similar manifold
of $k$-dimensional planes by integration over the set of all $j$-planes that
meet a given $k$-plane at a right angle. The case $j=0$ gives the classical
Radon-John $k$-plane transform. For any $j$ and $k$, our transform has a mixed
structure combining the $k$-plane transform and the dual $j$-plane transform.
The main results include action of such transforms on rotation invariant
functions, sharp existence conditions, intertwining properties, connection with
Riesz potentials and inversion formulas in a large class of functions. The
consideration is inspired by the previous works of F. Gonzalez and S. Helgason
who studied the case $j+k=n-1$, $n$ odd, on smooth compactly supported
functions.
