  The ground-state degeneracy and magnetocaloric effect in the spin-$1/2$
Ising-Heisenberg double-tetrahedral chain are exactly investigated. It is
demonstrated that the zero-temperature phase diagram involves two classical and
two quantum chiral phases with distinct degrees of the macroscopic degeneracy.
Different macroscopic degeneracies observed in the latter phases and at
individual ground-state phase transitions are confirmed by multiple-peak
dependencies of the specific heat and entropy on the magnetic field. The
cooling capability of the model is well illustrated by the magnetic-field
variations of the isothermal entropy change, temperature isotherms and the
magnetic Gr\"uneisen parameter.
