Can you act as a white-label payment service built on top of a payment processor?

Can a company who has a merchant account act as a white-label "middle-man" between another merchant and a payment processing company like Paypal? For example, a white-label company offers an API into which a customer-facing businesses can integrate. The customer-facing business then uses the API to manage and charge credit-card transactions made by their customers.

I've heard that this might be against the policies of Mastercard/Visa because they want the person with the merchant services account to be the actual merchant that the end user is facing.

I've looked into Chargify.com (a SaaS offering that manages recurring payments) to see how they operate and they don't act as a merchant. Rather, they allow you (the customer-facing site) to provide your own merchant account information to them, which they use to run the transactions for you. I wonder if a restriction of some sort is the reason they went with this model instead of using their own merchant account.

I'll probably call Paypal and a few other payment processors next week to ask them directly, but I was curious if anybody on here knew or had any experience in this area.