  Due to the presence of a nontrivial three-dimensional spin-orbital
entanglement, Sr$_2$RuO$_4$ may be a time-reversal invariant nematic $p$-wave
superconductor with coexisting in-plane and out-of-plane pairings. Here we
discuss various signatures of such a state if the out-of-plane pairing
component is dominant. First, the enhancement of the superconducting $T_c$
under in-plane uniaxial strains is nearly a quadratic function of the strain,
because the out-of-plane pairing lacks a linear-order coupling to the strain.
Second, when the strain applies along a certain in-plane direction, the nematic
$p$-wave pairing exhibits only a single phase transition as the temperature is
lowered. These are consistent with several recent uniaxial strain measurements,
which are otherwise hard to reconcile with chiral $p$-wave order. We further
show that the nematic $p$-wave state can be distinguished from the chiral
$p$-wave state through the velocity jumps of certain sound waves at the onset
of superconductivity. Possible implications for $\mu$SR experiment under strain
and NMR Knight shift measurement are also discussed.
