  Strontium ruthenate (Sr$_2$RuO$_4$) has long been thought to host a
spin-triplet chiral $p$-wave superconducting state. However, the singletlike
response observed in recent spin-susceptibility measurements casts serious
doubts on this pairing state. Together with the evidence for broken
time-reversal symmetry and a jump in the shear modulus $c_{66}$ at the
superconducting transition temperature, the available experiments point towards
an even-parity chiral superconductor with $k_z(k_x\pm ik_y)$-like $E_g$
symmetry, which has consistently been dismissed based on the
quasi-two-dimensional electronic structure of Sr$_2$RuO$_4$. Here, we show how
the orbital degree of freedom can encode the two-component nature of the $E_g$
order parameter, allowing for a local orbital-antisymmetric spin-triplet state
that can be stabilized by on-site Hund's coupling. We find that this exotic
$E_g$ state can be energetically stable once a complete, realistic
three-dimensional model is considered, within which momentum-dependent
spin-orbit coupling terms are key. This state naturally gives rise to
Bogoliubov Fermi surfaces.
