No, you solder it in the same way for both sides.

The “flipping” is handled mostly internally. For The right hand side, the firmware just maps the rows and columns differently than for the left hand side. For pins where this is not possible (Vcc, Gnd,…), there are jumpers on the other side that need to be soldered.

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looks great! what did you use to design the case and and how did you make this rough surface finish? It looks very good on the outside and will be good at hiding scraches!