Strassen (Strassen, J. Reine Angew. Math., 375/376, 1987) introduced the subrank of a tensor as a natural extension of matrix rank to tensors. Subrank measures the largest diagonal tensor that can be obtained by applying linear operations to the different indices (legs) of the tensor (just like the matrix rank measures the largest diagonal matrix that can be obtained using row and column operations). Motivated by problems in combinatorics and complexity theory we introduce the new notion of symmetric subrank of tensors by restricting these linear operations to be the same for each index. We prove precise relations and separations between subrank and symmetric subrank. We prove that for symmetric tensors the subrank and the symmetric subrank are asymptotically equal. This proves the asymptotic subrank analogon of a conjecture known as Comon's conjecture in the theory of tensors. This result allows us to prove a strong connection between the general and symmetric version of an asymptotic duality theorem of Strassen. We introduce a representation-theoretic method to asymptotically bound the symmetric subrank called the symmetric quantum functional in analogy with the quantum functionals (Christandl, Vrana, Zuiddam, J. Amer. Math. Soc., 2021), and we study the relations between these functionals.