  This paper uses Gaussian mixture techniques to dissect the Milky Way (MW)
stellar halo in angular momentum space. Application to a catalogue of 5389
stars near the plane of the Sagittarius (Sgr) stream with full 6D phase-space
coordinates supplied by Gaia EDR3 and SEGUE returns four independent dynamical
components. The broadest and most populated corresponds to the `smooth' MW
halo. The narrowest and faintest contains 40 stars of the Orphan stream. We
find a component with little or no angular momentum likely associated with the
GSE substructure. We also identify 925 stars and 7 Globular Clusters with
probabilities $>90\%$ to be members of the Sgr stream. Comparison against
$N$-body models shows that some of these members trace the continuation of the
leading/trailing tails in the Southern/Northern hemispheres. The new detections
span $\sim 800^\circ$ on the sky, thus wrapping the Galaxy {\it twice}.
