Two new negative-parity bands have been identified in the odd-odd $^{104}$Rh nucleus. According to their experimentally observed properties, they have the same $\pi {\left(1g_{9/2}\right)}^{-1}\otimes \nu {\left(1h_{11/2}\right)}^1$ high-j configuration as the previously known negative-parity chiral doublet bands. This observation raises the possibility of the existence of multiple chiral doublet bands, MχD, in this nucleus. Comparing the properties of the observed bands with results of detailed theoretical calculations, one can conclude that the lower energy parts of bands 1 and 2 are chiral partner bands with the $\pi {\left(1g_{9/2}\right)}^{-1}\otimes \nu {\left(1h_{11/2}\right)}^1$ two-quasiparticle configuration, while bands 3 and 4 are chiral partner bands with the $\pi {\left(1g_{9/2}\right)}^{-1}\otimes \nu {\left(1g_{7/2}\right)}^{-2}{\left(1h_{11/2}\right)}^1$ four-quasiparticle configuration. Thus, MχD based on different configurations is observed in $^{104}$Rh.