  In this paper, the biderivations without the skew-symmetric condition of the
twisted Heisenberg-Virasoro algebra are presented. We find some non-inner and
non-skew-symmetric biderivations. As applications, the characterizations of the
forms of linear commuting maps and the commutative post-Lie algebra structures
on the twisted Heisenberg-Virasoro algebra are given. It also is proved that
every biderivation of the graded twisted Heisenberg-Virasoro left-symmetric
algebra is trivial.
