It seems likely to me that there may be several convergent roots of the name. However, there are two probable: Celtic - and Spanish!
It seems fairly clear that an early written form of the name was Torrens - that being the best phonetic spelling of the name as written in Latin texts. To ask 'what is the original spelling?' is to miss the point. Remember that in those very few people could even write, let alone spell so the written spellings that occur were scribes' renderings of the name, as pronounced. There is a gravestone in the Vow graveyard on the Bann river in Ulster which bears the inscription 'Here lyeth the body of Huey Tornce who departed this life July 22 1712'. He is probably my own GGGGGGGrandfather!
The earliest records we have of the name are the story of Robert the Bruce's boatmen, in 1306. This story (but it may well just be that - a fictional story) links the Torrens name to the extreme southwest peninsula of Scotland, Mull of Kintyre and to the Island of Rathlin off the north east tip of Ireland.
It is also a historical fact that the early Scoti (from whom Scotland was named) were originally Irish and gradually moved into the area around this channel between Ireland and Scotland and then spread into Scotland. So it looks like the name Torrens may have originated in Ireland and not in Scotland.
Additional facts to support this are that the earliest recorded references to Scottish Torrances are around Avondale, circa 1500. Avondale is the valley of Avon Water which runs from east to west, draining into the Irish sea at Ayr.
A list of surnames common in Ireland at end of 16th Century (which was part of World Gen Web Project) referred to manuscripts kept at the Trinity College Library in Dublin, Ireland. On the list is Torran! This could easily evolve to Torrens.
Off the Rodd of Mull, in this area we find Torran Rocks. These are named, I am told from the gaelic 'torrann' - thunder; so maybe that's a parallel root of the name? Also relevant may be Torrin, on the Isle of Skye.
So it seems quite probable that our earliest ancestors were Irish!
The modern families of the name seem mostly to originate from the Avon Dale, radiating out from there, some to Edinburgh, some southwards into England. The best documented branches left Scotland for Ireland and thence to the new world. But other branches departed from Scotland to other parts of the world such as Canada, Australia and New Zealand. There is one line which is traceable back to a Thomas Torrence who resided in Dungiven, Londonderry and who was reputedly born in Scotland but who had served as an officer in the Swedish army before settling in Ireland.
Many Torrrens researchers notice the 'Torrens Triangle' (see Places) and assume this demarcates the area of land which tradition states was given to the brothers by Bruce in gratitude for saving his life. It has been noted by Professor Thomas F Torrance of Edinburgh, that there are two farms near Hamilton whose records are said to go back to the 9th century that are still being farmed by Torrances (letter dated January 1797): Quarter and Cruikedstane Farms.
I note Crookedstone on the OS map, 2m N.W. of Stonehouse and Quarter about 1m N.E. of this - both in the Avon Dale! These are a little south of the southern tip of the Torrence triangle.
There may also be at least one totally separate origin of the name Torrens: this is said to have arisen in the Basque area on the Spanish/French border and is said to have derived from torrentes - a torrent. However statistics I have seen indicate that this is a small subsection of the name, and does not trace back far in time. It is known that there were Scottish soldiers in many parts of the world around 1500-1600 and there seems a distinct possibility that this line also is in fact another branch of our Scottish family! I should be pleased to get any more information on this.
It should also be made clear that the spelling of the name really has no significance whatsoever, until perhaps around 1850. Before that, few people could even write, so in most cases the name had no spelling! This applies of course to all names and the variation of spelling in the Bann Valley visitation lists is interesting.
However the Torrens variant seems to be less common that -ance and -ence forms and it also seems that most of the Torrens families radiate via Ireland: notable lines are those from the Bann Valley (some of whose American lines are -ance and -ence), the Dungiven line, the Letterkenny line, and the line from David Torrens and siblings who, whilst of Scottish origin, migrated via Ireland.
There seems to a slight tendency for -ence to be of Scottish origin, and -ance of Irish.
The following lists the name variations (sorted in alpha sequence) of which I am aware. The modern versions which occur in charted family trees are emboldened. Of course, some of these may also derive from sources other than the Torrenses! For instance, Terrence probably (also) has a different source. But then I suspect even the name Torrens may have several sources!
Some believe Dorrens is also a variation. It is not!