The English term for "permuta" in the context of an artist or celebrity promoting a brand in exchange for products or services is "barter" or "in-kind sponsorship".
Here’s an example using "barter":
* "The celebrity agreed to promote the new skincare line on her social media in exchange for a year’s supply of the products as part of a barter deal."
Here’s an example using "in-kind sponsorship":
* "The musician received free instruments from the company as part of an in-kind sponsorship, where he mentioned the brand in his concert tours and social media posts."
Both terms convey the idea of exchanging promotion for goods or services without involving cash.