Symmetric C* property What can be said about elements $a$ of a C*-algebra which fulfil the 'symmetric C* property'
$$\| a^\ast a+ aa^\ast\|=2\| a\|^2$$

I'd guess that this is not a general property, but I don't have a good idea which elements (except for self-adjoint and normal ones, of course) should satisfy this.
If you assume that $a^*a$, $a a^\ast$ are compact (this is just to avoid working with continuous spectrum), the equality above holds iff there is an eigenvalue $\lambda$ with maximal norm $| \lambda | = \|a\|^2$ such that the eigenspaces of $\lambda$ for $a^\ast a$ and $a a^\ast$ have nontrivial intersection. That already give you that the condition is pretty rare. 

Indeed, two projections $p,q \in \mathcal{K}(\ell^2)$ satisfy that there is a partial isometry $a$ such that $p = a^\ast a$ and $q = a a^\ast$ iff $\dim(p\ell^2) = \dim(q \ell^2)$. But you can take $p \perp q$ and in that case:
$$
\| a^\ast a + a a^\ast \| = \max \{ \| a^\ast a \|, \| a a^\ast\| \} = 1.
$$
If you assume that $a^*a$, $a a^\ast$ are compact (this is just to avoid working with continuous spectrum), the equality above holds iff there is an eigenvalue $\lambda$ with maximal norm $| \lambda | = \|a\|^2$ such that the eigenspaces of $\lambda$ for $a^\ast a$ and $a a^\ast$ have nontrivial intersection. That already give you that the condition is pretty rare. 

Indeed, two projections $p,q \in \mathcal{K}(\ell^2)$ satisfy that there is a partial isometry $a$ such that $p = a^\ast a$ and $q = a a^\ast$ iff $\dim(p\ell^2) = \dim(q \ell^2)$. But you can take $p \perp q$ and in that case:
$$
\| a^\ast a + a a^\ast \| = \max \{ \| a^\ast a \|, \| a a^\ast\| \} = 1.
$$