  The Yukawa interactions of the SO(10) GUT with fermions in 16-plets (as well
as with singlets) have certain intrinsic ("built-in") symmetries which do not
depend on the model parameters. Thus, the symmetric Yukawa interactions of the
10 and 126 dimensional Higgses have intrinsic discrete $Z_2\times Z_2$
symmetries, while the antisymmetric Yukawa interactions of the 120 dimensional
Higgs have a continuous SU(2) symmetry. The couplings of SO(10) singlet
fermions with fermionic 16-plets have $U(1)^3$ symmetry. We consider a
possibility that some elements of these intrinsic symmetries are the residual
symmetries, which originate from the (spontaneous) breaking of a larger
symmetry group $G_f$. Such an embedding leads to the determination of certain
elements of the relative mixing matrix $U$ between the matrices of Yukawa
couplings $Y_{10}$, $Y_{126}$, $Y_{120}$, and consequently, to restrictions of
masses and mixings of quarks and leptons. We explore the consequences of such
embedding using the symmetry group conditions. We show how unitarity emerges
from group properties and obtain the conditions it imposes on the parameters of
embedding. We find that in some cases the predicted values of elements of $U$
are compatible with the existing data fits. In the supersymmetric version of
SO(10) such results are renormalization group invariant.
