I don't know about whatever company you've been hearing about, but--sure--it's possible to do.
The question, really, is: What is your intent? What's your goal? What would you like to accomplish from the transaction?
I'm not a lawyer or an accountant, so this isn't legal or accounting advice. Please consult those professionals as appropriate.
The answer to your question really depends on how the documents are constructed.
Keep in mind, though, that the investor or company you deal with is going to want to make some money, too. So while it's understandable that you'd like to protect yourself in case of a major market shift, so would the investor. If the market goes up 30%, the investor would like some of the profit. And if the market goes down 30%, the investor would like to spread some of the risk.
You can't have the best of all worlds: Moderate rent on your terms in case of a stable market, plus protection against any market shifts. Still, it all depends on the way the transaction is constructed.
You also run into a problem in that the cost of ownership in most areas is higher than the cost of rent. Put another way, if an investor bought your property, it's likely his cost would be higher than he could charge you in rent. And I'm guessing you'd like to reduce your monthly expenses. If you had equity in your property (or, put another way, if you owed less on your property than it was currently worth), you'd have a better chance at reducing your monthly payments.
In that case, you could do what's called a "subject to." You'd deed the property to the new owner subject to the existing mortgage. The old mortgage would stay in place (violating the lender's due on sale clause), and the new owner would be responsible for paying it. The new owner could then rent the property to you. The risk to you is that the new owner would fail to make the monthly mortgage payments.
It'd be far less desirable if the new owner had to go out and get a new mortgage. The additional costs involved would often make the transaction undesirable. And if the home is worth less than your mortgage, then you'd be faced with a short sale.
You could also use a land trust to accomplish the same thing: You'd transfer the property into your land trust. You'd be the beneficiary of the trust. Then you'd add the investor to the trust. The owner would be the trustee. You'd pay your rent to the trustee every month. That'd protect you more than a "subject to." However, the question still is: What is your goal in such a transaction?
As a renter, you protect yourself by having a long-term lease. Or a moderate-term lease with right of renewal. For example, you could have a 3-year lease with the right to renew it for up to 7 additional years in one-year increments. That'd protect you, but frankly wouldn't be too attractive to most investors. It would lock the investor in, not permitting him to take a profit from the sale of the property.
Again: What's your goal in such a transaction? It's a lot easier to tailor a solution when we know what you want to accomplish.
Hope that helps.