We study the problem of automated hypersafety verification of infinite-state recursive programs. We propose an infinite class of product programs, specifically designed with recursion in mind, that reduce the hypersafety verification of a recursive program to standard safety verification. For this, we combine insights from language theory and concurrency theory to propose an algorithmic solution for constructing an infinite class of recursive product programs. One key insight is that, using the simple theory of visibly pushdown languages, one can maintain the recursive structure of syntactic program alignments which is vital to constructing a new product program that can be viewed as a classic recursive program — that is, one that can be executed on a single stack. Another key insight is that techniques from concurrency theory can be generalized to help define product programs based on the view that the parallel composition of individual recursive programs includes all possible alignments from which a sound set of alignments that faithfully preserve the satisfaction of the hypersafety property can be selected. On the practical side, we formulate a family of parametric canonical product constructions that are intuitive to programmers and can be used as building blocks to specify recursive product programs for the purpose of relational and hypersafety verification, with the idea that the right product program can be verified automatically using existing techniques. We demonstrate the effectiveness of these techniques through an implementation and highly promising experimental results.