If so, this would suggest that only a settled population would be likely to build ceremonial monuments that would require a big investment of time and manpower. As yet, there is no evidence that Homo Sapiens Sapiens adopted an agrarian lifestyle that would lend itself to such "permanent" monument-building until much later than the periods we would need to be talking about for the Neanderthals to have done something similar.
This suggests (although obviously it doesn't prove or disprove) that megalithic monuments are associated with a settled community, which itself requires a non-migratory means of sustaining itself, i.e. farming. If so, unless evidence comes to light that the Neanderthals adopted farming techniques millenia before HSS did, it seems unlikely that they would have spent their energies on building permanent megalithic structures, even if they had the capacity or technology to do so, as it would be inconsistent with a largely migratory/seasonal way of life.
...stone-walled windbreaks, stone-walled residential buildings, stone foundations for houses and shelters, hunting hides, food storage buildings, as well as storage sites for sacred objects.
Engineering works of stone were also known, such as marine and freshwater fish traps, canals, ovens, protective coverings for sacred objects and path liners. There were also stone layouts on the ground of geometric and abstract design that had spiritual significance.
See Australia: The Land Where Time Began.
It’s dangerous to draw parallels with Australian aborigines and Neanderthals of course but it is interesting. Cave painting, as tiompan and others have alluded to, might have been a common activity and perhaps also ‘stone layouts on the ground of geometric and abstract design with spiritual significance’.