Air carrier Captain reported auto throttles advancing after touchdown. Captain held throttles and applied reverse thrust and brakes without further incident.

Narrative

Line check flight. LCA (Line Check Airman) in observer's seat. Entire flight and approach was uneventful; I disconnected the autopilot and was hand flying the approach. Touchdown was firm; on centerline; and in the touchdown zone; with thrust levers at or very near idle. On landing the speedbrakes auto deployed normally. Shortly thereafter the speedbrakes retracted and the aircraft lightened up" but no one thought it was a bounced landing in the debrief. The FO made the proper speedbrake calls and upon doing so I removed my hands from the thrust levers; that were at idle; to manually deploy the speedbrake. As I grabbed ahold of the speedbrake lever the thrust levers began to advance and I immediately closed them with my forearm while continuing to deploy the speed brake. Once the speed brake was deployed I slid my forearm off the thrust levers right to left to regain a hand grip on them. As I slid my forearm off the right thrust lever first; it was free to advance first; followed by the left trust lever a moment later following the full removal of my arm. This was relatively quick as I knew I had to catch the advancing throttles and I did not have a hand free as of yet. I held the thrust levers firmly to idle and applied thrust reverse. I applied brakes and continued the landing rollout normally without further incident. I do not recall any asymmetric reverse thrust. Pitch waivered a bit as the speedbrakes deployed; retracted; and deployed. As I recall the aircraft centerline was not an issue just assuring the nose didn't land hard on the runway was the goal. The discrepancy between the crew members began at the gate during debriefing. The LCA was unsure of what happed and initiated the debrief with an honest "What happened?". I explained that I did not know for sure; or the root cause; but explained that when the speedbrakes retracted I immediately went to deploy them again; then when the thrust levers advanced I immediately brought them back to idle. For the tired state I was in at the time; I felt it was an expeditious remedy to the situation; it was all I could do once it was set in motion; maintain control. Realize this entire scenario lasted only seconds; if I had to venture a guess I'd say three seconds. As fast as I could move my hands thrust lever to speedbrake; back to thrust levers; then apply reverse; was the timeframe. The discrepancy between crew recollection of events began at the gate during debrief. The first response from the LCA was that he felt I must have caused an inadvertent go around activation by hitting the go around switches. I politely disagreed; he tried one more time and I disagreed again. At that point he abandoned that idea and we began a lengthy discussion on what the root cause was. None of us knew for sure but we had suggestions that the LCA looked up on his iPad in an attempt to come up with a plausible explanation. The crew asked me if I had disconnected the autothrottles and in all honesty I could not recall. I had to review the audible auto throttle disconnect alarm and asked if we should have gotten one on touchdown; no one knew for sure at that time; and if we were in fact supposed to get one at some point (thrust rev activation?); that possibly we didn't; and missed the cue. We came up with the most plausible explanation being that I must not have disconnected the auto throttle prior to touchdown. I don't know for sure; but seemed plausible. We did agree we didn't think it was a bounced landing; but that the plane may have rebounded just enough to retract the speed brake and the thrust advanced momentarily as I removed my hand that was holding them at idle. The bigger discrepancy in aircraft state came from the LCA's comments in the Line Check Form. The LCA stated that "He grabbed the left throttle; pulled it to idle + reverse while right was still at approx 1/2 throttle; he then pulled both to idle + reverse." The LCA thought that having one thrust lever in reverse; and oneat 1/2 forward thrust was a dangerous aircraft state and failed me for the line check. I would agree if it happened. I have no recollection of this happening; and recall no yawing moment; I have no idea why I would ever retard one lever and pull only one into reverse. What I believe happened (and I can be wrong) was the LCA just saw the momentary split between thrust levers advancing as I removed my forearm; and saw me closing each thrust lever. I was under the impression that both thrust levers had to be at idle; and gear tilt; to have either thrust lever go into reverse. So I'm not entirely sure if the scenario is possible; hopefully the aircraft data will shed some light on this. Cause: If I had indeed forgot to disconnect the auto throttles prior landing; it was already too late once the scenario began to play out. Had the speedbrake not retracted I would have easily been able to simply disconnect the autothrottles with my hand remaining in control of the thrust levers. Once my hand left the thrust levers for the speedbrake there was nothing I could do but attempt to quickly manage each control individually as quickly as possible. I was doing the best I could at the time. Now when I state that it was the "best I could do at the time"; please reference my two previous reports regarding my hotels and resulting poor rest. My rest was interrupted at one hotel as I was escorted out of my room; I finished out my rest in the ZZZ airport without accommodations getting ready in the men's room of the lounge. That same night I was left with finding a hotel on my own after being sent to one with a known lack of vacancy. I found myself worked up; frustrated; and writing a report at night; not sleeping well; and did not eat. The final night before this flight I show up in ZZZ1 tired and hungry but unable to sleep early enough in the day for adequate rest before my showtime. I did not eat a meal from the time I was escorted out of my ZZZ hotel until dinner in ZZZ1 the following day. I was "leaning forward" doing my best to get these flights done for the company in what was a difficult time due to the winter storm that crippled the aviation industry. I also had only flown one leg on Day 18 since my last Check Ride on Day 0; so I knew I would be rusty and wanted to fly a couple legs before a line check. I should have never accepted a line check without flying previous legs; that was entirely my fault. I knew better.Contributing factors were definitely fatigue on my part and I believe the FOs part as well. The FO did bring up the fact it was a late night flight/fatigue in our brief. By the amount of missed radio calls he made; and ever increasing through the flight; it was obvious fatigue was setting in. The LCA had to prompt him many times for missed radio calls or other small missed items. Another factor was the LCA may not have been in the best frame of mind; before we got to the hotel van on departure he informed us of the urgency of getting this flight out expeditiously so he could catch a quick connecting flight. The LCA informed us that his family member was found in need of urgent medical care and was hospitalized. So of course there was some pressure to perform in an expeditious manner. I feel a review of the debrief conversation would definitely show how fatigued the flight crew was as we fumbled for an explanation of what unfolded; it was probably embarrassing. In the end we really don't know what happened; but just built a plausible explanation after the fact. No one caught the issue in flight or on touchdown. Suggestions: Clearly addressing fatigue would be a mitigation well worth considering. I believe we all did well and did quite well as a crew complement trapping errors and working together; sadly we (me) may have missed one small item that had a well executed flight fall to pieces in the last few seconds. Keeping pilots flying more regularly is where I would like to see improvement. It's hard to fly only one leg a month then drop a pilot into the aftermath of a crippling winterstorm scenario; fatigued; and expect perfection. I can honestly say that I showed up prepared; with the best rest I could get; ready to operate safely. As the flight delayed and the morning drew closer; I guess I dropped the ball on remembering to disconnect the autothrottle. I'm glad I decided to skip my rest and fill out this narrative instead; because my hotel room phone just rang and would have disturbed my rest. It was the front desk; the associate wanted to reach out to tell me that she's sorry; and that she knows I'm in the room; but the company insists I'm not; and wants the hotel to release the room for another employee to use. This is the every day company fatigue pilots endure that wears on us. We show up ready; and with all due respect; many of us feel aggravated that we don't have the support; as if no one is listening to us. Overall improvement in pilot moral would probably mitigate a lot of operational and safety related issues. I am not just pointing the finger; I can assure you I am not pleased with myself; my performance; or the outcome of this event."