Already solved by Charlie and Steven Lord, but I proceeded with a computer solution and got the same answer:  40.

----------  Here is my Python code:  ---------

/* (since lists in Python are zero based, I defined locations as:

012

345

678

and thus the neighbor locations for location 0, for example, are locations 1 and 3.) */

----------  

from itertools import permutations

digits = [1,2,3,4,5,6,7,8,9]

nbr_locs = [[1,3],[0,2,4],[1,5],[0,4,6],[1,3,5,7],[2,4,8],[3,7],[4,6,8],[5,7]]

def isAdjacent(aPerm):

    for n in range(1,9):

        loc = aPerm.index(n)

        other_locs = nbr_locs[loc]

        adj = False

        for otloc in other_locs:

            if aPerm[otloc] == n+1:

                adj = True

        if not adj:

            return False

    return True

goodOnes = []

countall = 0

countNbr = 0

for perm in permutations(digits):

    countall += 1

    if isAdjacent(perm):

        countNbr += 1

        goodOnes.append(perm)

        

print(countall)

print(countNbr)

for t in goodOnes:

    a=100*t[0]+10*t[1]+t[2]

    b=100*t[3]+10*t[4]+t[5]

    c=100*t[6]+10*t[7]+t[8]

    print('',a,'\n',b,'\n',c)

    print('')

------------------------

Since any solution is also a solution if rotated 90 degrees (4 ways), or flipped over (2 ways), at first I thought there should be 5 solutions each with 8 different equivalent rotations/reflections.  In fact there are only three patterns which I call the "S", the "G", and the "R".  There are 8 instances of the S pattern, and 16 each of the other 2 patterns.  The reason is there is one more degree of freedom with the latter two patterns:  repeating the pattern going from 1 up to 9 is different than the same pattern going from 9 down to 1.  

S 8

G 16

R 16

Technically, I guess you could subdivide "G" and "R" into G up, G down, R up, and R down and thus have 5 patterns each with 8 reflections/rotations.

=======================

123 S

654

789

123 R

874

965

123 G

894

765

129 R

438

567

145 R

236

987

167 S

258

349

187 G

296

345

189 R

276

345

321 S

456

987

321 R

478

569

321 G

498

567

329 G

418

567

345 G

216

987

345 R

276

189

345 G

296

187

349 S

258

167

541 R

632

789

543 G

612

789

543 R

672

981

543 G

692

781

567 G

418

329

567 R

438

129

567 G

498

321

569 R

478

321

761 S

852

943

765 G

814

923

765 R

834

921

765 G

894

123

781 G

692

543

789 G

612

543

789 R

632

541

789 S

654

123

921 R

834

765

923 G

814

765

943 S

852

761

965 R

874

123

981 R

672

543

987 G

216

345

987 R

236

145

987 S

456

321