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The involutory birack counting invariant is an integer-valued invariant of unoriented tangles defined by counting homomorphisms from the fundamental involutory birack of the tangle to a finite involutory birack over a set of framings modulo the birack rank of the labeling birack. In this first of an anticipated series of several papers, we enhance the involutory birack counting invariant with quantum weights, which may be understood as tangle functors of involutory birack-labeled unoriented tangles.