Crystal Systems vs. Symmetry in Elasticity

[Facial]

Here I go with another question:

Does a monoclinic (Bravais) crystal system necessarily reflect a monoclinic elastic symmetry in elasticity?

I was skimming over a few paragraphs from my old composites book in college, and the author gives feldspar (no specific mineral, just the mineral family here) as an example of a 'monoclinic' material in the context of elasticity. I am a bit skeptical whether the crystal system actually corresponds one-on-one with mechanical behavior. Can anyone clarify this?