What was the structural basis for the classical symphony?

The sonata form.

In particular, the sonata form typically was the structure of the first movement of classical symphonies. Sonata form follows this pattern:

Exposition: Two musical themes are set forth or "exposed" to the listeners. (Thus the label "exposition.") The first theme is usually strong, the second quieter, more gentle.

Development: Then the two themes are explored or "developed" in various ways and played off of one another.

Recapitulation: Finally, the themes are restated in the order in which they were introduced in the exposition section and reaffirmed as the musical themes of the movement.