  [Background] Various cluster states of astrophysical interest are expected to
exist in the excited states of $^{28}{\rm Si}$. However, they have not been
identified firmly, because of the experimental and theoretical difficulties.
[Purpose] To establish the $^{24}$Mg+$\alpha$, $^{16}$O+$^{12}$C and
$^{20}$Ne+2$\alpha$ cluster bands, we theoretically search for the
negative-parity cluster bands that are paired with the positive-parity bands to
constitute the inversion doublets. We also offer the isoscalar monopole and
dipole transitions as a promising probe for the clustering. We numerically show
that these transition strengths from the ground state to the cluster states are
very enhanced. [Method] The antisymmetrized molecular dynamics with Gogny D1S
effective interaction is employed to calculate the excited states of $^{28}{\rm
Si}$. The isoscalar monopole and dipole transition strengths are directly
evaluated from wave functions of the ground and excited states. [Results]
Negative-parity bands having $^{24}$Mg+$\alpha$ and $^{16}$O+$^{12}$C cluster
configurations are obtained in addition to the newly calculated
$^{20}$Ne+2$\alpha$ cluster bands. All of them are paired with the
corresponding positive-parity bands to constitute the inversion doublets with
various cluster configurations. The calculation show that the band-head of the
$^{24}$Mg+$\alpha$ and $^{20}$Ne+2$\alpha$ cluster bands are strongly excited
by the isoscalar monopole and dipole transitions. [Conclusions] The present
calculation suggests the existence of the inversion doublets with the
$^{24}$Mg+$\alpha$, $^{16}$O+$^{12}$C and $^{20}$Ne+2$\alpha$
configurations.Because of the enhanced transition strengths, we offer the
isoscalar monopole and dipole transitions as good probe for the
$^{24}$Mg+$\alpha$ and $^{20}$Ne+2$\alpha$ cluster bands.
