In terms of facade greening, much of what characterised the Baroque period continued on into the Classical era... above all the further development of small espalier gardens and Talut walls, most of which were probably small. New climbing plants were introduced or bred. In garden art this era is also known as the "Biedermeier period." But with Romanticism and its veneration of the 'castle,' there was a contradicting impulse: instead of order, light, and austerity, ivy-covered and seemingly ancient walls were revered.
There are some cities which had a lively architectural development in the Classical period. Some of the many greenings produced in these cities are even still visible from that time; for example, in Berlin, Potsdam, Weimar, Eutin, and Putbus. See the photo examples below in the gallery. (In classical garden art, strict and ordered elements predominated, similar to the models from antiquity: stone structures-- pyramids and temples, staircases and balustrades. There was virtually no place for greenery.)
The "Biedermeier" era, from approx. 1815 to 1850, was a lively style within Classicism. It turned away from fanatical idealism of any kind. Instead, it was all about enjoying life and living in style with family and friends. Garden art played a significant role, and the "Biedermeier-Gardens" -- with flowers, wall greening, and espalier fruit-- emerged.