  The exact nature of the unconventional superconductivity in Sr$_2$RuO$_4$
remains a mystery. At the phenomenological level, no superconducting order
parameter proposed thus far seems able to coherently account for all of the
essential experimental signatures. One prominent observation is the polar Kerr
effect, which implies an anomalous Hall effect. Without appealing to extrinsic
factors originating from crystalline imperfection, this peculiar effect may be
interpreted as an evidence of a bulk chiral Cooper pairing with nonzero orbital
angular momentum, such as $p+ip$ or $d+id$. In this paper, we propose
alternative possibilities with complex mixtures of distinct helical p-wave
order parameters, namely, $A_{1u}+iA_{2u}$ and $B_{1u}+iB_{2u}$ in the group
theory nomenclature. These states essentially consist of two copies of
inequivalent chiral p-wave pairings with opposite chirality and different
pairing amplitudes, and therefore support intrinsic Hall and Kerr effects. We
further show that these states exhibit salient features which connect to
several other key observations in this material, including the absence of
spontaneous current, a substantial Knight shift drop, and possibly signatures
in uniaxial strain and ultrasound measurements.
