From 巻(ま)き (maki, 連(れん)用(よう)形(けい) (ren'yōkei, “stem or continuative form”) of the verb 巻(ま)く (maku)) + 貝(かい) (kai). The kai changes to gai as an instance of rendaku (連濁).
まきがい • (makigai) ←まきがひ (makigafi)?