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This may be technically true, but I think it is a little misleading.
To consider a wave "extraordinary" it must have at least some
component of its E-field parallel to the optic axis. (I am restricting
my consideration to uniaxial substances like quartz.) If a wave
propagates along the optic axis, then both linear polarizations are
necessarily identical in every respect.
Consider a quartz plate whose optical axes are
(?? I assume you mean to say "whose optical axis is")
also normal to its flat surfaces. ...