  To clarify the geometric information encoded in the $SO(D+1)$ spin-network
states for the higher dimensional loop quantum gravity, we generalize the
twisted-geometry parametrization of the $SU(2)$ phase space for $(1+3)$
dimensional loop quantum gravity to that of the $SO(D+1)$ phase space for the
all-dimensional case. The Poisson structure in terms of the twisted geometric
variables suggests a new gauge reduction procedure, with respect to the
discretized Gaussian and simplicity constraints governing the kinematics of the
theory. Endowed with the geometric meaning via the parametrization, our
reduction procedure serves to identify proper gauge freedom associated with the
anomalous discretized simplicity constraints and subsequently leads to the
desired classical state space of the (twisted) discrete ADM data.
