Abstract Two sets of oscillations in the magnetoreflection of high purity α-Sn have been observed. These are well explained by the ‘inverted’ band structure. The Γ 8 + − Γ 7 − energy separation is found to be 0.413 eV, and independent of temperature over the range 1.5°–85°K. Other directly determined quantities are the band-edge masses m ∗ c (Γ 8 +) = 0.028 m, m ∗ v (Γ 8 + = −0.195 m, and m ∗ v (Γ 7 −) = −0.058 m. the k . p interaction parameter E p and the valence-band spin-orbit splitting Δ are deduced to be 24.0 and 0.80 eV, respectively. The results of this and other experiments raise doubts about ‘current thoughts’ on the positions of the Γ 15 conduction band and Γ 15 valence band. There is a discrepancy between interband and intraband effective masses, which is also characteristic of InSb.