For C*-algebras $A$ and $B$, we construct three related classes of cross-norm completions $A\otimes_{\mu} B$ of $A\odot B$. The first class of these constructions produces Banach algebras, the second Banach $*$-algebras, and the third produces C*-algebras. For certain discrete groups $G_1$ and $G_2$, such as noncommutative free groups, our C*-completions of $C^*_r(G_1)\odot C^*_r(G_2)$ coincide with a Brown-Guentner type C*-completion of $\ell^1(G_1\times G_2)$ and produces $2^{\aleph_0}$ distinct C*-completions of $C^*_r(G_1)\odot C^*_r(G_2)$. A Banach $*$-algebra $A$ is symmetric if the spectrum $\sigma(a^*a)$ is contained in $[0,\infty)$ for every $a\in A$ and rigidly symmetric if the Banach space tensor product $A\otimes_\gamma B$ is symmetric for every C*-algebra $B$. A theorem of Kügler asserts that every type I C*-algebra is rigidly symmetric. Leveraging our new constructions, we establish the converse of Kügler's theorem by showing for C*-algebras $A$ and $B$ that $A\otimes_\gamma B$ is symmetric if and only if $A$ or $B$ is type I. This strongly settles a question of Leptin and Poguntke from 1979.