  Combined constraints from the CDF and D0 Collaborations on models of the
Higgs boson with exotic spin $J$ and parity $P$ are presented and compared with
results obtained assuming the standard model value $J^P=0^+$. Both
collaborations analyzed approximately 10~fb$^{-1}$ of proton-antiproton
collisions with a center-of-mass energy of 1.96 TeV collected at the Fermilab
Tevatron. Two models predicting exotic Higgs bosons with $J^P=0^-$ and
$J^P=2^+$ are tested. The kinematic properties of exotic Higgs boson production
in association with a vector boson differ from those predicted for the standard
model Higgs boson. Upper limits at the 95% credibility level on the production
rates of the exotic Higgs bosons, expressed as fractions of the standard model
Higgs boson production rate, are set at 0.36 for both the $J^P=0^-$ hypothesis
and the $J^P=2^+$ hypothesis. If the production rate times the branching ratio
to a bottom-antibottom pair is the same as that predicted for the standard
model Higgs boson, then the exotic bosons are excluded with significances of
5.0 standard deviations and 4.9 standard deviations for the $J^P=0^-$ and
$J^P=2^+$ hypotheses, respectively.
