For a recent portfolio review I made "leave behind" cards printed with my Epson 3800 on Ilford Galerie Gold Fibre Silk. This paper is thick (~320 gsm) and smooth, with an almost indiscernible stipple on its warm, glossy surface.
This paper would make lovely bizcards, albeit expensive ones. Essentially, my 6x6 inch leave-behinds were supersized biz cards.
Harman Glossy FB Al is smoother and glossier, and even more expensive, as I recall.
Oops, I just reread your post and see you need paper printable on both sides. Scratch the two I mentioned. You may have a hard time finding double-sided glossy paper, since glossy finishes wouldn't likely survive then trauma of being the backside of a sheet as it passes thru the printer.