PA28-160 Flight Instructor reported a rough running engine on climb out resulted in a return to a safe landing.

Narrative

Aircraft X set out for a training flight. A CFI and student pilot onboard. All manufacturer's checklists were appropriately followed throughout all phases of flight. Student pilot conducted a thorough passenger briefing describing his role and the instructor's role. Student pilot conducted a thorough takeoff briefing to include all contingencies. Upon takeoff abeam the numbers; both pilots felt a rough engine. The pilots had a small window to turn back to the Runway. Pilots elected to turn towards the runway due to the low altitude and that being about the only emergency landing site. ATC was very busy and crew did not have time to notify before turning inbound. Crew still determined it was appropriate to head towards the airport for a landing. When there was a break on the radio; crew immediately jumped in and told ATC returning for landing due to rough engine. ATC was very helpful and cleared the way. Before landing checks were completed and crew safely landed on the Runway. Engine parameters did appear normal throughout the taxi; run-up and flight. Crew taxi'd aircraft to our mechanic for further investigation. Crew felt situation was appropriately and safely followed per aircraft manuals and ATC procedures. Crew always remains open for further discussion. Situation with student was thoroughly debriefed as a great learning event.