  The quantum steering ellipsoid inscribed inside the Bloch sphere offers an
elegant geometric visualization of two-qubit states shared between Alice and
Bob. The set of Bloch vectors of Bob's qubit, steered by Alice via all possible
local measurements on her qubit, constitutes the steering ellipsoid. The
steering ellipsoids are shown to be effective in capturing quantum correlation
properties, such as monogamy, exhibited by entangled multiqubit systems. We
focus here on the canonical ellipsoids of two-qubit states realized by
incorporating optimal local filtering operations by Alice and Bob on their
respective qubits. Based on these canonical forms we show that the reduced
two-qubit states drawn from pure entangled three-qubit permutation symmetric
states, which are inequivalent under stochastic local operations and classcial
communication (SLOCC), carry distinct geometric signatures. We provide detailed
analysis of the SLOCC canonical forms and the associated steering ellipsoids of
the reduced two-qubit states extracted from entangled three-qubit pure
symmetric states: We arrive at (i) a prolate spheroid centered at the origin of
the Bloch sphere -- with longest semiaxis along the z-direction (symmetry axis
of the spheroid) equal to 1 -- in the case of pure symmetric three-qubit states
constructed by permutation of 3 distinct spinors and (ii) an oblate spheroid
centered at $(0,0,1/2)$ inside the Bloch sphere, with fixed semiaxes lengths
(1/Sqrt[2],\, 1/Sqrt[2],\, 1/2)), when the three-qubit pure state is
constructed via symmetrization of 2 distinct spinors. We also explore volume
monogamy relations formulated in terms of the volumes of the steering
ellipsoids of the SLOCC inequivalent pure entangled three-qubit symmetric
states.
