(Python)
import itertools
n=3 #replace with desired value of n
# Here we list all sequences of successive differences that give rise to rotational row complete latin squares.  See
good_list=[]
for i in itertools.permutations([i for i in range(1,2*n-1)]):
....entries=[0]
....val=0
....for j in range(n-1):
........val=(val+i[j])%2*n
........if val not in entries:
............entries.append(val)
....if len(entries)==2*n:
........good_list.append(list(i))
#here we define the Gilbert transformations.  See E. N. Gilbert, ``Latin squares which contain no repeated digrams", SIAM Review, vol. 7, no. 2, 1965, page 197.
def R(ourlist):
....newlist=[]
....for i in range(len(ourlist)):
........newlist.append(2*n-(ourlist[-(i+1)])%(2*n))
....return(newlist)
def T(ourlist):
....j=ourlist.index(int(n))
....newlist=ourlist[j+1:]+[int(n)]+ourlist[:j]
....return(newlist)
def M(sig,ourlist):
....newlist=[]
....for i in range(len(ourlist)):
........newlist.append(ourlist[i]*sig%(2*n))
....return(newlist)
phivals=[]
for i in range(2,(2*n)):
....if gcd(i,2*n)==1:
........phivals.append(i)
#Here we sort sequences into equivalence classes and count.
rawlist=copy(good_list)
seenlist=[]
classes=[]
for trylist in rawlist:
....if trylist not in seenlist:
........equivclass=[trylist]
........t=T(trylist)
........if t not in equivclass:
............equivclass.append(t)
............seenlist.append(t)
........r=R(trylist)
........if r not in equivclass:
............equivclass.append(r)
............seenlist.append(r)
........rt=T(r)
........if rt not in equivclass:
............equivclass.append(rt)
............seenlist.append(rt)
........for phi in phivals:
............mphi=M(phi,trylist)
............if mphi not in equivclass:
................equivclass.append(mphi)
................seenlist.append(mphi)
............rmphi=R(mphi)
............if rmphi not in equivclass:
................equivclass.append(rmphi)
................seenlist.append(rmphi)
............tmphi=T(mphi)
............if tmphi not in equivclass:
................equivclass.append(tmphi)
................seenlist.append(tmphi)
............rtmphi=R(tmphi)
............if rtmphi not in equivclass:
................equivclass.append(rtmphi)
................seenlist.append(rtmphi)
........classes.append(equivclass)
len(classes)