Wrapping paper cannot be recycled if it contains sparkles, glitter, sequins, aluminum foil, artificial texture, adhesive gift tags, or plastic. Nor can it be recycled if it has been laminated or if it still has lots of excess tape, tapes or ribbons attached. Yes, you can recycle Christmas wrapping paper unless it's metallic, has glitter or has velvet flocks. Not all types of gift wrapping are recyclable, but many are.
Regular gift-wrapping paper can be recycled, but generally not effectively. Some local recyclers accept it in recycling bins. Others offer seasonal collection events and many request that it simply be thrown away. It's OK to leave adhesive tape on the wrapping paper when you recycle it.
Furoshiki, traditional Japanese cloths used to wrap gifts or products, come in several sizes and are a great way to give as a gift.