= Integral generator signs constrain Steenrod comparisons
Suppose two integral cohomology groups of a space are free of rank one. An integral isomorphism can send each chosen generator only to itself or its negative. After reduction modulo $p$, naturality of a <Steenrod reduced power> between those groups therefore permits its coefficient to change only by a sign, even though an abstract vector-space change of basis could rescale by any nonzero field element. At $p=5$, coefficients one and two cannot be related by signs, so they obstruct an integral <homotopy equivalence>. This constraint is stronger than comparing the isolated mod-five Steenrod modules with arbitrary bases.
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