The effective mass model of the electronic structure and intersubband absorption in p-doped twinning superlattices is devised. As the bulk states basis, heavy hole (HH), light hole (LH) and split-off (SO) band Bloch functions, properly rotated according to the crystal orientation, are used. Furthermore, delta potential is introduced at the interface between two layers to model the corresponding microscopic potential. Peculiar electronic structure offers coupling between otherwise forbidden states in composite superlattices. This coupling arises from the change of both the off-diagonal terms of the velocity operator and the Luttinger parameters across two interfaces belonging to the superlattice period. The peak of the absorption coefficient for x polarized light (z is the growth direction) arises mainly from transitions between LH1 and HH2 miniband. This peak is located in the midwavelength infrared window for all three investigated semiconductors, GaAs, Ge, and Si. It is shown that the dipole matrix elements are responsible for absorption. It is important to note that the magnitude of the absorption coefficient agrees with the results of the pseudopotential theory. The results indicate the usefulness of this structure, which has recently been realized, in the state-of-art quantum well infrared photodetectors.