An antonym for astride are:

astraddle

athwart

on the back of

piggyback

sitting on

straddling

Lying or being across, or in a crosswise direction; athwart; -- often opposed to longitudinal., Anything that is transverse or athwart., The longer, or transverse, axis of an ellipse., To overturn; to change., To change from prose into verse, or from verse into prose.

Probably corruption of 'athwart', meaning sideways. After all, forward & backward, so why not athward?

Situated or placed across something else; transverse; oblique., Fig.: Perverse; crossgrained., Thwartly; obliquely; transversely; athwart., Across; athwart., A seat in an open boat reaching from one side to the other, or athwart the boat., To move across or counter to; to cross; as, an arrow thwarts the air., To cross, as a purpose; to oppose; to run counter to; to contravene; hence, to frustrate or defeat., To move or go in an oblique or crosswise manner., Hence, to be in opposition; to clash.