We derive the relativistic factor for splitting of the $g$-factors of a fermion and its anti-fermion partner, which is important for placing constraints on dimension-5, $CPT$-odd and Lorentz-invariance-violating interactions from experiments performed in a cyclotron. From existing data, we extract limits (1$��$) on the coupling strengths of the temporal component, $f^0$, of a background field (including the field amplitude), which is responsible for such $g$-factor splitting, with an electron, proton, and muon: $|f^0_e|< 2.3 \times 10^{-12} ~��_{\textrm{B}}$, $|f^0_p|< 4 \times 10^{-9} ~��_{\textrm{B}}$, and $|f^0_��|< 8 \times 10^{-11} ~��_{\textrm{B}}$, respectively, in the laboratory frame. From existing data, we also extract limits on the coupling strengths of the spatial components, $d^{\perp}$, of related dimension-5 interactions of a background field with an electron, proton, neutron, and muon: $| {d}_e^{\perp} | \lesssim 10^{-9} ~��_{\textrm{B}}$, $| {d}_p^{\perp} | \lesssim 10^{-9} ~��_{\textrm{B}}$, $| {d}_n^{\perp} | \lesssim 10^{-10} ~��_{\textrm{B}}$, and $| {d}_��^{\perp} | \lesssim 10^{-9} ~��_{\textrm{B}}$, respectively, in the laboratory frame.

6 pages. Minor corrections and new references added