From the Super Mario Wiki
The Mirror is a Brothers Bear item that appears in Donkey Kong Country 3: Dixie Kong's Double Trouble!. It is required to complete a trade and has a second, optional use for completing a puzzle.
The Mirror is sold at Bazaar's General Store at a price of 50 Bear Coins. According to the shop owner, Bazaar, the Mirror was formerly in possession of Cranky Kong's great grandfather. If Dixie and Kiddy Kong purchase the Mirror, they can trade it to Barter for his No. 6 Wrench, which they can in turn give to Björn for him to repair his chairlifts.
Baffle is also in need of the Mirror, which he needs to decode a message. When the Kongs provide him the Mirror, Baffle can decode the message, which reveals the location of KAOS Kore's Banana Bird Cave. Providing the Mirror is optional because the Kongs can still find the cave without talking to him.
The Mirror has a unique mirror-type on each Brothers Bear while they are in possession of it. If Barter has the Mirror, the layout of his swap shop is inverted; this gimmick is absent from the Game Boy Advance port. If Baffle owns the Mirror, he speaks in reverse. Dixie and Kiddy can reacquire the Mirror from either Brothers Bear by paying ten Bear Coins to them. The Mirror is the only Brothers Bear item that the Kongs can keep in their inventory up until they complete their journey.