[math-fun] Would someone with a clearer head please explain why

these are unequal: {Sin[InverseFunction[(π + #1) + Sin[π + #1] &][t]], Sin[InverseFunction[#1 + Sin[#1] &][π + t]]} (real t) while these aren't? {Sin[InverseFunction[(π + #1) + Sin[π + #1] &][π - t]], Sin[InverseFunction[#1 + Sin[#1] &][π + t]]} —rwg