In this talk we will review definitions and properties of double-slice genus and equivariant 4-genus. Combining these ideas, we will define a notion of equivariant double-slice genus for strongly invertible knots and cover a variety of results differentiating the equivariant double-slice genus from other genera. Using the equivariant double-slice genus we begin to study symmetric surfaces in $S^4$ up to equivariant isotopy, constructing interesting examples of symmetric surfaces in $S^4$ and introducing notions of equivariant stabilization distance for symmetric surfaces.

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