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[Background] Various cluster states of astrophysical interest are expected toexist in the excited states of $^{28}{\rm Si}$. However, they have not beenidentified 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 thenegative-parity cluster bands that are paired with the positive-parity bands toconstitute the inversion doublets. We also offer the isoscalar monopole anddipole transitions as a promising probe for the clustering. We numerically showthat these transition strengths from the ground state to the cluster states arevery enhanced. [Method] The antisymmetrized molecular dynamics with Gogny D1Seffective interaction is employed to calculate the excited states of $^{28}{\rmSi}$. The isoscalar monopole and dipole transition strengths are directlyevaluated from wave functions of the ground and excited states. [Results]Negative-parity bands having $^{24}$Mg+$\alpha$ and $^{16}$O+$^{12}$C clusterconfigurations are obtained in addition to the newly calculated$^{20}$Ne+2$\alpha$ cluster bands. All of them are paired with thecorresponding positive-parity bands to constitute the inversion doublets withvarious cluster configurations. The calculation show that the band-head of the$^{24}$Mg+$\alpha$ and $^{20}$Ne+2$\alpha$ cluster bands are strongly excitedby the isoscalar monopole and dipole transitions. [Conclusions] The presentcalculation 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 theisoscalar monopole and dipole transitions as good probe for the$^{24}$Mg+$\alpha$ and $^{20}$Ne+2$\alpha$ cluster bands.

Inversion doublets of reflection-asymmetric clustering in 28Si and their isoscalar monopole and dipole transitions