Here's a very simple way to obtain $(E \wedge E)^{\Bbb Z_2}$ without having to resort to representations (at least if $E$ is connective). Consider the functor from spectra to spectra given by $$E \mapsto \Sigma^\infty \Omega^\infty E .$$ That is, the suspension spectrum of the zeroth space of $E$.
Let $$E \mapsto P_2(E)$$ be its quadratic approximation in the sense of Goodwillie's calculus of homotopy functors (that is, the second stage of the Goodwillie tower).
Then $P_2(E)$ has the homotopy type of $(E \wedge E)^{\Bbb Z_2}$ whenever $E$ is connective.
Incidentally, one can also see that this comes equipped with a fiber sequence $$ D_2(E) \to P_2(E) \to E $$ which amounts the tom Dieck's splitting in the case when $E$ is a suspension spectrum. Here, $D_2(E) = (E\wedge E)_{h\Bbb Z_2}$ is the quadratic construction. In general, this sequence needn't split.