The Lot of Fortune is found by measuring the distance from the Sun to the Moon and projecting it from the Ascendant. That is the day formula and nobody disputes it. The question is what to do with a chart in which the Sun is below the horizon.
The reversing position. Take the distance from the Moon to the Sun instead, and project that. The lot is then always measured from the luminary that rules the sect, the Sun by day, the Moon by night, and Fortune and the Lot of Spirit swap formulae with each other, staying mirror images across the Ascendant in both sects. This is what Vettius Valens does, what Paulus Alexandrinus sets out, and what the Arabic and much of the medieval Latin tradition inherited. It is also the position of the twentieth-century Hellenistic revival: the Project Hindsight translations from the 1990s brought the Greek material back into circulation, and Chris Brennan, Demetra George and Robert Hand all work with the reversal.
The non-reversing position. Ptolemy gives a single formula in the Tetrabiblos and applies it to both sects, saying explicitly that he does this. He also does not use the Lot of Spirit, so the symmetry argument does not arise for him. William Lilly follows Ptolemy in Christian Astrology, and because Lilly is the fountainhead of the English tradition, three centuries of English-language astrology, including most twentieth-century textbooks and a great deal of astrological software still shipping today, calculate Fortune without reversing it.
Why it is not a small thing. In a night chart, the two methods place Fortune at points reflected about the Ascendant. That is generally a different sign, usually a different house, and often the opposite side of the chart. Since the whole use of the lot is to read its house and sign, the two methods do not produce a slightly different reading. They produce a different one.
Where the argument currently stands. The historical question is close to settled: the weight of the surviving Greek evidence is that reversal was standard, and Ptolemy was the outlier rather than the norm. Ptolemy's enormous influence on the later transmission is the reason his version dominated for so long. The practical question is not settled at all, because "what the majority of ancient sources did" and "what produces readings a working astrologer finds useful" are different tests, and practitioners who have tried both do not all report the same result.
This site stores the day formula, which both camps agree on, and flags the night calculation as contested rather than picking a side.