This is Octagonal Hex, based on Eight-Sided Hex, which appears in an article by Larry Back in the Spring 2001 issue of Abstract games magazine. Mr. Back used a regular rhombus board shape, but such a grand conception deserves a side for each border row.

The object here is to connect two pairs of your sides. One side may be part of both pairs, or the pairs may be distinct. For example, if your border rows are labeled circularly A, B, C, D, you may connect A to B and A to C. Another possibility is to connect A to B and C to D. These connecting paths may connect to each other, although they do not have to.
This is Square Hex, which appears in the same article. I believe it also appears in Ea Ea's book "Mudcrack Y and Poly-Y." The central cell has just four neighbors, rendering it less important. The shortest path connecting your sides runs along either of your opponent's border rows.

One possible disadvantage both these boards may have is, by equalizing the importance of different regions of the board, you reduce the effectiveness of the swap rule. What's the point of a swap rule, if EVERY initial move should be swapped? Perhaps a more complicated opening protocol, such as three-move equalization, would help, but adding more fiddly rules detracts from the purity and simplicity of the concept behind Hex.
Thanks to Ed Collins for clueing me in about these variants. He has his own page about them here.

Abstract Games magazine has a website
here.

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