Light aircraft Flight Instructor reported experiencing an NMAC with another light aircraft in the pattern at HRU airport.

Narrative

My student and I were conducting landing practice at HRU. We made radio calls throughout our traffic pattern practice; as well as initially upon entry to the airport. Based on the windsock at the airport when we arrived; we chose to use Runway 35 (HRU has no METAR). We landed twice at the airport with no incidents and were on our third touch-and-go upwind of Runway 35 when I spotted traffic.I took the controls from my student and took evasive action to offset to the right of incoming traffic. I made a call on the radio that I was offsetting to the right for traffic. The other traffic made no response. I queried the other aircraft if they could hear us and if they had heard any of our prior radio calls. No response. I called and asked if traffic in the pattern had made any calls prior to entering the airport. The other aircraft responded and said they had made a 4 mile final call. Neither my student nor I had heard this call. We had been listening to traffic calls from other aircraft on the shared frequency at other airports. Later; another pilot who had been in the traffic pattern at K78 (airport north of HRU that shares the same frequency) said that he had heard my traffic pattern calls at HRU without any issue.The other pilot who nearly collided with us made no effort to deviate from their final approach course into HRU despite the near miss. They made radio calls after landing; one to state that they were taxiing clear of the runway and then another that they were taxiing to park. These two calls; in addition to their response to me about having made a four mile radio call; were the only three calls I heard from the pilot.I did not catch the full tail number of the pilot and my student and I decided to leave the airport. Because our near-collision was so close; I could see into the other plane and it appeared that there had only been one person in the left seat.The other plane had no ADS-B out and I was unable to find any record of traffic at the airport besides myself on FlightRadar or FlightAware. There was also no indication of the other aircraft on our Multi-function Flight Display (MFD) or traffic map; or on either my student or mine's iPads.We did have a TCAS alert in our plane; which only sounded after I had already spotted the plane and taken evasive action. My student and I were able to dodge the other plane due to the evasive action I took to turn us to the right. Had I not done so; it is highly likely that we would have collided fully head-on or at least by clipping wings.The other plane should have also turned to the right upon our near-miss; but they continued straight into the airport and had only responded after 3 inquiries to them on frequency.Based on winds; which were relatively calm; either runway would have been equally possible. However; the other plane also should have deferred to existing traffic patterns/runway in use as my student and I had been in the pattern for the last 10 minutes and had made radio calls on each leg.