I think I just found the first reference to forward crossovers in the skating literature.
In 1788, Gerhard Ulrich Anton Vieth gave a lecture to a group of (what he called) friends. That lecture was published in Neue Litteratur und Völkerkunde and, in 1790, reprinted as a book with the title Über das Schrittschuhlaufen. It’s a little book that was lost to skating historians for quite a while, in part because people kept getting the title wrong by substituting Schlittschuhlaufen (the current German word for skating) for Schrittschuhlaufen. In the 18th century, Schrittschuhlaufen was the correct term for skating. But that’s a topic for another time. Both versions are now readily available in Google books.
Here’s the crossover:
And here’s my (rough) translation of it:
On the contrary, if something of the centripetal force still remains after the centrifugal force has already become equal to zero, then this results either in anxious and uncomfortable gripping with the hands, to find a stable point for the center of gravity, or, in those who know how to help themselves a little better, stepping over the skating foot with the free foot, which is also not entirely compatible with beauty, whereby the remaining centripetal force is suddenly and somewhat violently made equal to zero. Such sudden transitions are never graceful, and in this case there is still a danger of getting the skates entangled when stepping over, in which case the whole body plays a very poor role.
It seems that forward crossovers were originally mistakes made by beginners trying to keep their balance while skating.
G. U. A. Vieth. 1790. Über das Schrittschuhlaufen. Graz: mit v. Widmanstättenschen Schriften.