Nice find. It looks like what is generically known as a 'handkerchief clip' but, depending on the design and construction, these were also used to hold gloves and sometimes chiffon or silk scarves.
Again depending on the construction, they may have been worn on the wrist, attached to a purse (small handbag) or to a chatelaine. In olden times, some were also made with the clip attached by a short chain to a finger ring so that a small lace handkerchief was easily to hand. That may sound unhygienic to have it flapping around like that, but a ladies fine handkerchief wouldn't have been used for 'nose-blowing'. It was more of a pretentious fashion accessory, would often have been sprayed with perfume and dabbed on the cheeks or brow when a flush came on; or used for both flirtation and modesty/concealment depending on mood/circumstances (a bit like the way a fan would be used).
Those specifically made for gloves may have the arms of the clasp in the form of two gloved hands.
