To clear things a little bit here. A Router is not a Controller, It’s a way to define a client-side route map (similar to Rails’s routes.rb). This helps routing client-side pages to certain actions/handlers. And that’s different from a controller’s job which is to provide a bit of orchestration between Models and Views. And there is actually more than one way to do this using Backbone. Quoting from Backbone’s documentation:
References between Models and Views can be handled several ways. Some
people like to have direct pointers, where views correspond 1:1 with
models (model.view and view.model). Others prefer to have intermediate
“controller” objects that orchestrate the creation and organization of
views into a hierarchy. Others still prefer the evented approach, and
always fire events instead of calling methods directly. All of these
styles work well.
This brings three different approaches to accomplish this. The first one is pretty straightforward which is to have the model object included as a property to the view.
The second one proposes including a third component that performs this role of orchestration. I believe this can be helpful in quite large and complex applications. For this I encourage you to look at Chaplin, a sample application architecture using Backbone.js. The guys have done a great job in separating things out and also introduced the concept of a Controller into the architecture.
The last approach is suggesting using events to mark for actions and mediator to handle these actions. For this I encourage you to look into the mediator and Publish/Subscribe JavaScript patterns.