Abstract We study formalisms for temporal and spatial reasoning in the modern, algebraic and modeltheoretic, context of infinite-domain Constraint Satisfaction Problems (CSPs). We show how questions on the complexity of their subclasses can be solved using existing results via the powerful use of primitive positive (pp) interpretations and pphomotopy. We demonstrate the methodology by giving a full complexity classification of all constraint languages that are first-order definable in Allen’s Interval Algebra and contain the basic relations (s) and (f). In the case of the Rectangle Algebra we answer in the affirmative the old open question as to whether ORD-Horn is a maximally tractable subset among the (disjunctive, binary) relations. We then generalise our results for the Rectangle Algebra to the r-dimensional Block Algebra.