Both of these URLs will render the same results:
example.com/brands/adidas/shopby/mens
example.com/mens/shopby/adidas
These combinations are possible because there is a /brand and /mens entry point, then the layered navigation (amasty) will allow different combinations. Ideally, these 2 URLs shouldn't exist, I'd like to have only 1 version of this specific set of combinations for SEO purposes.
How do you do this?
Note: We are using Magento 1 with Amasty layered navigation
Possible solution:
By adding a "new page" in the layered navigation, I can control the canonical URL, so when bots land on example.com/brands/adidas/shopby/mens or example.com/mens/shopby/adidas, I can choose to rel canonical to example.com/brands/adidas/shopby/mens and therefore solve the problem. This will work, but it's not scaleable as it needs manual configuration and very prone to human error.
Realize that this not a technical issue related, it is a content issue.
You are replicating the catalog in distinct categories because you have categories that are attributes and vice versa. If you don't want to change the catalog you can remove one of those categories in robots.txt
Disallow: /brands/adidas/shopby/*
Cheers