We propose and demonstrate a short and broadband silicon mode-conversion polarization splitter-rotator (PSR) consisting of a mode-conversion taper and an adiabatic coupler-based mode sorter both optimized by adiabaticity engineering (AE). AE is used to optimize the distribution of adiabaticity parameter over the length of the PSR, providing shortcut to adiabaticity at a shorter device length. The total length of the PSR is 85 µm. The design is compatible with standard silicon photonics platforms and requires only one patterning step. Fabricated PSR has a polarization cross talk of less than -20 dB over the entire O-band for the TE polarization and a polarization cross talk of less than -15 dB from 1267 to 1348 nm for the TM polarization. Overall, the PSR shows low polarization cross talk (-15 dB) over a bandwidth of 81 nm in the O-band. Cross-wafer measurements show that the PSR has good fabrication tolerance.