The Diagonal style is a phase of Gothic architecture. The style appears to have been characterized by a predominance of random, slanted lines in various angles and lengths. The Thirteenth Century additions to the Cathedral of Saint Gorboduc included a number of "cross-purposed groining, crockets, finials, and imitation cane rood screens" that were thought to be of this style (Saint Fidgeta and Other Parodies; 34).
In English Gothic architecture there is a style and period known as perpendicular, so named for its emphasis on vertical lines. Bellairs spoofs this with a style more chaotic and susceptible to internal conflicts - something that easily describes the jumbled assembly of Saint Gorboduc's.
Cross purpose: A contrary or conflicting purpose or understanding, especially an unintentional or misconceived one.