Correlations are symmetric, but is evidence for may not be (depending on how you interpret the phrase): P(A|B) ≠ P(B|A) (unless P(A) == P(B)).
However, the likelihood ratio (P(B|A)/P(B|~A)), a.k.a., the quantity you actually care about when updating on new evidence, is symmetric.
Correlations are symmetric, but is evidence for may not be (depending on how you interpret the phrase): P(A|B) ≠ P(B|A) (unless P(A) == P(B)).
However, the likelihood ratio (P(B|A)/P(B|~A)), a.k.a., the quantity you actually care about when updating on new evidence, is symmetric.