cyclic node renumbering (for each element) gives different "eparts"

This is to report an anomaly. I hope it is my error (if so kindly point it out).

meshfile.dat (with just 9 quad elements):
9
1 2 6 5
2 3 7 6
3 4 8 7
5 6 10 9
6 7 11 10
7 8 12 11
9 10 14 13
10 11 15 14
11 12 16 15

If I use the binary mpmetis.exe to make two partitions:
mpmetis.exe meshfile.dat 2

I get the following eparts data (which seems to be bad):
0 1 0 1 0 0 1 1 1

If I use the METIS API function METIS_PartMeshDual I get a similar result if ncommon=1. If ncommon=2 (which is more appropriate) I get the eparts data (again bad) as:
0 0 1 1 0 0 1 1 1

However, using a cyclic renumbering of nodes within each element gives an acceptable eparts (ncommon=2).
meshfile.dat (with just 9 quad elements):
9
2 6 5 1
3 7 6 2
4 8 7 3
6 10 9 5
7 11 10 6
8 12 11 7
10 14 13 9
11 15 14 10
12 16 15 11

eparts: 0 1 1 0 0 1 0 1 1

I wonder why does this happen?
I will be grateful for any feedback.