ASP Routing Be Used To Create "clean" URLs For .ashx Handlers?

I have some REST services using plain old IHttpHandlers. I'd like to generate cleaner URLs, so that I don't have the .ashx in the path. Is there a way to use ASP.NET routing to create routes that map to ashx handlers? I've seen these types of routes previously:

// Route to an aspx page
RouteTable.Routes.MapPageRoute("route-name",
"some/path/{arg}",
"~/Pages/SomePage.aspx");
// Route for a WCF service
RouteTable.Routes.Add(new ServiceRoute("Services/SomeService",
new WebServiceHostFactory(),
typeof(SomeService)));

Trying to use RouteTable.Routes.MapPageRoute() generates an error (that the handler does not derive from Page). System.Web.Routing.RouteBase only seems to have 2 derived classes: ServiceRoute for services, and DynamicDataRoute for MVC. I'm not sure what MapPageRoute() does (Reflector doesn't show the method body, it just shows "Performance critical to inline this type of method across NGen image boundaries"). I see that RouteBase is not sealed, and has a relatively simple interface:

public abstract RouteData GetRouteData(HttpContextBase httpContext);
public abstract VirtualPathData GetVirtualPath(RequestContext requestContext,
RouteValueDictionary values);

So perhaps I can make my own HttpHandlerRoute. I'll give that a shot, but if anyone knows of an existing or built-in way of mapping routes to IHttpHandlers, that would be great.