DDoS can be analysed simply by checking the volume and pattern of traffic hitting the router. Any decent router will have something called SNMP (Simple Network Management Protocol). SNMP captures information about the performance of the router (in this case, it can be used for all manner of devices) and allows an external service to take the data, aggregate and report on it. One of the advantages of SNMP is that is happens out-of-bounds of the actual traffic and therefore should not impact the performance of the router and its traffic. Ideally, SNMP traffic for a router should go over a completely separate LAN (a management LAN) to the data. SNMP can be a security risk of course and so needs to be handled carefully out at the network boundary.
Any enterprise grade router will have that capability. Many consumer grade routers also have it such as my Billion 7800N.
An alternative that captures more detail and may detect DDoS attacks that simple SNMP monitoring might miss requires something in the traffic stream to intercept the traffic and analyse it. This could happen external to the router by changing the DNS entry for the network to be monitored so that it passes through an external monitoring service which, in turn, forwards the data on to the router. You can do this inside the network boundary as well of course. The danger here is adding too much latency to the traffic so that it impacts performance.
I'm not so sure what you mean by your last question. It is feasible to route incoming traffic back out again rather than inwards but I'm not sure why you would want to. Certainly if the aim were to monitor traffic, this would be a very inefficient method.
UPDATE: From your comment regarding the anturis services.
Anturis is an example of set of products that cover a number of types.
In all of these cases, you have to take some kind of action to allow the monitoring to occur. Varying from installing software on your servers, through to reconfiguring your routers (to allow access to SNMP), through to allowing Ping from the Internet.