We combine the estimated metallicities [Fe/H], abundances [α/Fe], positions and motions of a sample of over 13 000 local (7 < R/kpc < 9, 0.5 < |z|/kpc < 2.5) Sloan Digital Sky Survey (SDSS)/Sloan Extension for Galactic Understanding and Exploration (SEGUE) G-type dwarf stars to investigate the chemo-orbital properties of the Milky Way's disc around the Sun. When we derive the orbital properties reflecting angular momentum, circularity and thickness as a function of [α/Fe] versus [Fe/H], we find that there is a smooth variation with [α/Fe], a proxy for age. At the same time, the orbital properties of the old stars with [α/Fe] ≳ 0.25 do show a transition with [Fe/H]: below [Fe/H] ≃ −0.6 the orbital angular momentum decreases, and the orbits become significantly non-circular and thicker. Radial migration of stars into the solar neighbourhood would naturally result in a smooth variation in the orbital properties, but the latter old metal-poor stars form a clear challenge, in particular because a basic feature of radial migration is that stars remain on near-circular orbits. When we next select stars on near-circular orbits, we indeed find besides the α-young ‘thin-disc’ stars a significant contribution to the α-old ‘thick-disc’ metal-rich stars. However, the remaining α-old ‘thick-disc’ stars on eccentric orbits, including nearly all old metal-poor stars, are difficult to explain with radial migration alone but might have formed through early-on gas-rich mergers. We thus find chemo-orbital evidence that the thicker component of the Milky Way's disc is not distinct from the thin component as expected from smooth internal evolution through radial migration, except for the old metal-poor stars with different orbital properties which could be part of a distinct thick-disc component formed through an external mechanism.