Cross-ply corrugated cardboard

Plywood is a laminate material made from several thin sheets of wood, glued together with the grain at different angles (typically alternating by 90 degrees). This makes the board stiff in both planar axes.

Corrugated cardboard is effectively three layers of kraft paper- two outer 'liners', with the central one pleated up and down to form 'flutes'. This makes it stronger, however it has a directionality; it is easier to bend and fold along the flutes.
Double-walled varieties have five layers; flat sheets alternating with corrugations, and higher multi-wall options all exist.

So, all multi-walled cardboard I've seen has the flutes in the same direction. This obviously is sub-optimal for many purposes, since there is increased flexibility and creasing in one orientation.
I suspect this is primarily due to ease of manufacture - corrugating machines are apparently enormous, and produce cardboard at a very high rate. I think that generating longitudinal pleats in a continuous manner may not be straightforward.
Nevertheless, even if this is not feasible I believe that a relatively simple engineering solution may be available. Corrugated cardboard with only two layers -one liner, one corrugated medium ('single face board') is available as an option at some manufacturers.
The strategy requires that we run two manufacturing lines - notionally but probably not literally at right angles to each other. One produces single-wall board, while the other produces single-face board. Potentially these operations may be performed sequentially on the same corrugator, and these intermediates rolled onto reels for storage.
In the final, assembly step, the top layer of the single- wall is glued to the exposed flutes of the single-face board at right angles to its own crenelations. Obviously after this operation the product will not bend well in either orientation and may have to be cut into sheets and stacked for storage rather than rolled up.