B-737 Captain reported a FMC malfunction which affected the autoflight system resulting in unreliable course guidance information on flight instrument displays during climb in mountainous terrain. The Captain maintained the course and did not follow the instrument course reversals.

Narrative

We were cleared for takeoff Runway XX. It was mostly still dark outside; but there was some hint of daylight peeking up over the mountains from the east. On climb out I followed the flight director and the course guidance as indicated on the FMC. The ZZZ procedure page shows to climb outbound on the XXX Radial; then when above X;X00 ft. make a left turn and intercept the XXX radial inbound to the ZZZ VOR. At approximately 6000' MSL I noticed the course reversal flip from a left turn; to a right turn on the EFIS. Essentially; instead of making a left turn it was commanding us to make a right turn. I double checked my course guidance; and I was dead on center line on the XXX radial outbound. There was a light 9-10 knot direct cross wind from the northeast; but it wasn't enough to really push us into the protected airspace west of the XXX radial. Within a second or two; I noticed the magenta line flip back to normal to command a left turn. I cross checked the FO (First Officer) side; and his side was also commanding a right turn; then switched to reflect a left turn. This flip flop back and forth from left turn to right turn happened five or six times in the course of 15 - 30 seconds. It finally landed on a right turn; which was opposite of the procedures page and prompted me to ask the FO to advise ATC that we were experiencing an FMC issue and to ask them if they had a preference on which way we turned. ATC came back and stated that it was pilot discretion. By the time we were at the point where we needed to start the turn I noticed daylight coming across the mountain tops off to my right (east); and darkness to my left (west). The winds were from the east at roughly 9 -10 knots; and I could easily see the silhouette from the top of the mountains. At this point the flight director and magenta line on the EFIS were commanding a right turn. The wind was blowing from east to west; so I opted to follow the flight director and turn into the wind direction which would ultimately tighten my turn radius; pitch to exchange some airspeed for altitude while further tightening up my turn radius; and fly the path that I could visibly see and avoid the mountains. I wasn't sure if the wind had us drifting left of course while outbound. If this were the case then I could theoretically understand that the FMC would try and keep you within the protected zone; and might try and have you turn into the direction of the CDI deflection if you were blown off course. However; we were wings level for the majority of the outbound leg; and the Flight Director didn't seem to indicate that we were drifting much if at all. The aircraft symbol identified that we were on the XXX radial (the magenta line) and that we had not drifted off course. There are no notes from the company that suggest FMC anomalies on this departure. I have never seen nor heard of the airplane commanding an opposite direction turn of this nature. I was always under the assumption that it would follow the published guidance as programmed off the Jepp diagram. Having it flip flop on us multiple times was a real cause for concern; especially if we were in IMC conditions while still in that valley with mountains on both sides. Someone at company needs to take a closer look at this scenario/issue and try and find a solution. Or maybe; this is normal and I have never seen it before now. Either way; everyone I have spoken with says it's not normal for the FMC to command multiple back and forth flip flops of a course reversal such as this occurrence.