If Buyer Agency or Seller Agency doesn't fit the bill, there are some lesser used forms that exist in some states and not in others:

No Agency - In this case, a buyer or seller, lessee or lessor, can elect to not be represented in an agency capacity at all. In agency parlance, they would be a "customer" rather than a "client". There is nothing wrong with entering into a transaction without agency representation. All licensees and especially all REALTORS are bound by law and by the REALTOR code to treat all parties to a transaction honestly, competently, and to provide full disclosure of any material facts about the property that they are aware of. Agency representation raises the bar, adding loyalty, confidentiality, and a few other good things into the equation. No agency, in a nutshell, has some good and bad depending on your attitude toward the process. Not being bound to an agent allows a buyer or seller more flexibility to "play the field" and work with more than one agent at the same time. The down side is that it cuts both ways. Agents operating without a contract tend to spend more time and effort on clients that are loyal and working with them exclusively for the obvious reasons. Some states only have No Agency.

Broker Agency - A broker's agent is just what it seems it would be. The broker is representing only themselves and their brokerage in a transaction. It is similar to "No Agency" in that the buyer or seller is not represented, but is a little more forthright in outlining the fact that the broker is acting in their own best interests. Brokers must still treat all concerned fairly and honestly, but are spending their time and energy in a way that benefits them, not at the expense of the buyer or seller, but without extra effort being expended to act in the direct interest of the parties to the transaction. Broker Agency is not allowed in some states.

Sub-Agency - A sub-agent is usually a broker from a company other than the listing agency who brings the buyer, but is still representing the seller as a sub-agent of the listing agent. So, the sub-agent is usually a seller's agent, but it is not as obvious to the public as when a consumer contacts the actual listing agent. Consequently, sub-agency has been outlawed in some states (e.g. Vermont) to avoid consumer confusion.

Potentially the most important part of agency is not the form that is selected by the consumer, but the proper disclosure of who is representing whom by the agent dealing with the public. In the interest of fair play and open business practice, agents must disclose who, if anyone, they are representing before any substantive contact with the uninformed consumer takes place (usually before a showing or a listing presentation).

Again, whole books have been written about the various forms of agency and the rationale behind them, as well as the legal and political ramifications of the subject. Obviously, the above is one agent's perspective on the subject and is not meant to be a legal interpretation or exhaustive explanation. A good source of additional and more detailed information on agency can be found at each state's real estate commission.