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Let $\mathcal{X}$ be an r.i. space over a non-atomic probability space. Let $\rho:\mathcal{X}\rightarrow(-\infty,\infty] $ be proper, convex, and increasing. It is known that order lower semicontinuity does not imply $\sigma(\mathcal{X},\mathcal{X}_n^sim)$ lower semicontinuity. However, if $\rho$ is additionally law invariant, then the implication does hold. This result indicates an interplay between law invariance and order. More surprisingly, we show that if $\rho$ is real-valued and law invariant, then both order and $\sigma(\mathcal{X},\mathcal{X}_n^\sim)$ lower semicontinuity is automatic at every $X\in\mathcal{X}$ such that $X^-\in\mathcal{X}^a$.