Some of my questions were answered when I discovered
"Public Sculpture of Greater Manchester" by Wyke and Cocks (a recent work) on the "Google books" page. A reproduction of the relief - a poor one - may be seen there, together with a good description of its history.
The relief at Manchester is dated 1886, is of marble, and measures 1.1 metres by 2.08 metres. The base of the curved seat is said to "carry a Greek inscription" but this is not evident in my copy of Hollyer's photo-graph, and we are not told what the inscription says! The condition of the work is described as "good"; but it does not look good, and we are not informed about where "good" stands on the scale. In numismatics "good" actually means "deplorably badly worn and almost unacceptable, even as a gap-filler," and the photo-graph looks just like that.
Bates completed a first version of the work, almost certainly in plaster, in 1883 while a student at the Royal Academy Schools, and received a gold medal and a travelling studentship of £200 for it. The work was exhibited at the Royal Academy in the following year, and reproduced in the Illustrated London News. Bates was then commissioned by the architect Alfred Waterhouse to produce a larger, marble version of the work for Owens College in Manchester. Waterhouse presented, and presumably paid for, the marble relief to the University. In 1886 this too was shown at the Royal Academy before being sent to Manchester, where it was installed on the rear wall of the Council Chamber.
"Socrates" continued to be greatly admired; Edmund Gosse confirmed its position as "one of the pre-eminent works of the New Sculpture." In 1900 Bates's widow - he seems to have been a hetero-sexualist which is disappointing - presented the initial plaster version to his old school, the Thomas Alleyne's Grammar at Stevenage.
The reproduction accompanying the "Google" text is as I say exceedingly poor:
but it does confirm for us that Socrates is on the right.
A further version, carved from wood, exists it seems:
Some Northern American "business" people are attempting to "flog" this rather ugly wooden version on the Inter-web without naming its creator or provenance, even.
The questions that remain then are:
1) why is the version in Roberts's painting back to front? Was it a different version made by Bates, or was it the whim of Roberts? (Misprints may now be ruled out.)
2) Are there any good modern photo-graphs of the work?
3) What does the Greek inscription say?
4) Is there a photo-graph of the plaster version at the School? - if indeed it remains there?