An involutory decomposition is a decomposition, due to an involution, of a
group into a twisted subgroup and a subgroup. We study unexpected links bet
ween twisted subgroups and gyrogroups. Twisted subgroups arise in the study
of problems in computational complexity. In contrast, gyrogroups are group
-like structures which first arose in the study of Einstein's velocity addi
tion in the special theory of relativity. In particular, we show that every
gyrogroup is a twisted subgroup and that, under general specified conditio
ns, twisted subgroups are gyro-commutative gyrogroups. Moreover, we show th
at gyrogroups abound in group theory and that they possess rich structure.