I was involved in the debugging of a similar situation to this....
The scenario was a laundrette with two of the machines using 3 phases motors. These had been installed & working fine for years but, one Monday morning, both machines would not spin but everything else worked fine. The washing machine tech was summoned but, after swapping PCBs & the motors, couldn't find make either of the machines spin - this is where I became involved.
The tech had done some basic measurements - Phase to earth for each of the 3 phases, found 230 ish volts and concluded that the supply was good .... he hadn't looked phase to phase. Checking the distribution board revealed something very similar to what the OP has seen .... L1 to L2 = 400V, L2 to L3 = 400V, L1 to L3 = 0V but L1 to E = 230V, L2 to E = 230V, L3 to E = 230V. Clearly the supply was not actually three phase .... but it must have been on Friday!
The DNO guys took a little convincing too, but eventually an older guy was found who had seen something similar before and knew where to look....
What had happened..... On Saturday a 'fault' had resulted in the blowing of the L1 & L3 substation fuses for DNO mains (630A fuses). The fuses were duly replaced by the on-call guy & held. Everything appeared to power up Ok and nobody reported being off supply - Job done! Well, not quite... it transpired that the motors in the laundrette were the only 3 phase loads on that feeder so the real cause went undetected. After a little searching & some digging, the real culprit was found....
A fault in an underground cable joint had resulted in a short L1 to L3, thus taking out two of the fuses. The energy of the fault had blown a break into L3 conductor but also welded the L3 & L1 conductors together on the load side of the fault. When the fuses were replaced, the 'fault' had apparently cleared - all 3 'phases' became live. Consumers before the fault had a true three phase supply (but no actual three phase loads) those after the fault really had L1, L2, L1, with the only true 3 phase load being the motors in the laundrette!