I am looking for a reference which would treat some very elementary properties of the principal symbols of pseudodifferential operators, such as conjugation and products. In particular, I am interested in seeing an exact proof which shows e.g. that either the principal symbol $\sigma_{A^*}$ of a conjugate of a pseudodifferential operator $A$ is equal to the conjugate of the principal symbol of $A$, $\sigma_A$ or that there $\sigma_{A^*A} = \sigma_{AA^*} = \left|\sigma_A\right|^2$. Googling has not helped me because I am so new to microlocal analysis/PsiOps that I don't know how to ask the right questions. As of writing, I couldn't find any source which proved/disproved these two properties.
Edit: I also do not know how to prove these claims.