  We performed comprehensive theoretical and experimental studies of the
electronic structure and the Fermi surface topology of two novel quantum
materials, MoSi$_2$ and WSi$_2$. The theoretical predictions of the electronic
structure in the vicinity of the Fermi level was verified experimentally by
thorough analysis of the observed quantum oscillations in both electrical
resistivity and magnetostriction. We established that the Fermi surface sheets
in MoSi$_2$ and WSi$_2$ consist of 3D dumbbell-shaped hole-like pockets and
rosette-shaped electron-like pockets, with nearly equal volumes. Based on this
finding, both materials were characterized as almost perfectly compensated
semimetals. In conjunction, the magnetoresistance attains giant values of
$10^4$ and $10^5\,\%$ for WSi$_2$ and MoSi$_2$, respectively. In turn, the
anisotropic magnetoresistance achieves $-95$ and $-98\,\%$ at $T=2\,$K and in
$B=14\,$T for WSi$_2$ and MoSi$_2$, respectively. Furthermore, for both
compounds we observed the Shoenberg effect in their Shubnikov-de Haas
oscillations that persisted at as high temperature as $T=25\,$K in MoSi$_2$ and
$T=12\,$K in WSi$_2$. In addition, we found for MoSi$_2$ a rarely observed
spin-zero phenomenon. Remarkably, the electronic structure calculations
revealed type-II Dirac cones located near 480 meV and 710 meV above the Fermi
level in MoSi$_2$ and WSi$_2$, respectively.
