An instructor pilot conducting training reported Tower issued last second go-around instructions which placed them head on to air carrier traffic resulting in an NMAC.

Narrative

Upon arriving to the terminal area from the southeast we were instructed to enter a left downwind 17R from about 15 miles out and then asked by tower to confirm instructions when closer. We continued to 17R and cleared to land at least 10 (miles) out. After entering the runway environment as instructed we flew a snug pattern almost directly over the west parallel. While rolling in to final at low level in full landing configuration we were then instructed to extend our downwind and the landing clearance was canceled. That was an extremely late call to do that action as we were lined up on 17R on a slight angle to final at that time. Not knowing what the issue was and the student not taking action I assumed flying and took the aircraft. I immediately executed a climb then rolled right to a northern direction then needed to take evasive action because of the head on with an airliner in my front windscreen. This was an extremely near miss and extremely late call by the tower to take any kind of action. Arrival time in the terminal area was about XA30 as my notes on landing time are missing. I am sure you can look it up and be much closer.If we were allowed to continue we would have been clear of the runway with plenty of time to allow the following aircraft to land with no issues. The other resolution would have been when we were still actually on downwind to have been given instructions to continue until base was called. Neither of us in the aircraft had a visual on the airliner until we were taking evasive action. We were never advised by tower that there was another aircraft on final all thought we did hear approach talking to others. This action of making an extremely aggressive maneuver could have been avoided as it turned out to be extremely dangerous for us at low level and steep angles while slow to avoid the other aircraft. I would clarify this as a serious incident.