The odd-$Z$ $^{251}$Md nucleus was studied using combined $gamma$-ray and conversion-electron in-beam spectroscopy. Besides the previously observed rotational band based on the $[521]1/2^-$ configuration, another rotational structure has been identified using $gamma$-$gamma$ coincidences. The use of electron spectroscopy allowed the rotational bands to be observed over a larger rotational frequency range. Using the transition intensities that depend on the gyromagnetic factor, a $[514]7/2^-$ single-particle configuration has been inferred for this band, i.e., the ground-state band. A physical background that dominates the electron spectrum with an intensity of $simeq$ 60% was well reproduced by simulating a set of unresolved excited bands. Moreover, a detailed analysis of the intensity profile as a function of the angular momentum provided a method for deriving the orbital gyromagnetic factor, namely $g_K = 0.69^{+0.19}_{-0.16}$ for the ground-state band. The odd-$Z$ $^{249}$Md was studied using $gamma$-ray in-beam spectroscopy. Evidence for octupole correlations resulting from the mixing of the $Delta l = Delta j = 3$ $[521]3/2^-$ and $[633]7/2^+$ Nilsson orbitals were found in both $^{249,251}$Md. A surprising similarity of the $^{251}$Md ground-state band transition energies with those of the excited band of $^{255}$Lr has been discussed in terms of identical bands. Skyrme-Hartree-Fock-Bogoliubov calculations were performed to investigate the origin of the similarities between these bands.
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