Lubos Motl makes an honest attempt to answer this question for me. His answer begins as follows:
[T]he Z-boson is identical to its antiparticle, so there are no two distinct varieties, only one Z_0 (with three polarizations i.e. spin states). It's also true for the neutral Goldstone/Higgs boson described [in the original post] which is eaten by the Z-boson: this particle is identical to its antiparticle, too.
Late he discusses Weyl neutrinos and their relevance to this point.
The premise that the Z-boson is identical to its antiparticle would seem to have to be true, if indeed, three weak force particles eating three of four Higgs bosons scenario is accurate. (Lubos believes that the world is supersymmetric, implying that there are more than four Higgs bosons in all, of which three are "eaten" by weak force bosons; I am agnostic on the matter and willing to wait and see if a Standard Model Higgs boson will be found at all.)
He's probably right. But, I'm not sure that a Z-boson is differently situated than a neutrino, which comes in two types (left handed neutrinos and right handed anti-neutrinos), because a Z-boson, as a weak force particle, only interacts with left handed parity particles and hence may not have both left parity and right parity varieties itself. Then again, perhaps parity does not make sense for bosons in the way that they do for fermions.
Lepton number conservation, violations of which have never been experimentally observed, seems to require that the distinction between a neutrino and anti-neutrino have some meaning. Indeed, the neutrino was predicted at all principally as a means of conserving lepton number before we knew that it had mass. But, if instead a neutrino and an anti-neutrino are identical, and hence the neutrino is a Majorana particle, then this distinction is somewhat hard to fathom. Of course, the Dirac v. Majorana particle status of the neutrino is unresolved although the failure to experiments thusfar to detect neutrinoless double beta decay, an expected experimental signature of Majorana neutrinos, increasingly bound Majorana neutrino scenarios.
Of course, if there was some genuine distinction between a Z boson and an anti-Z boson (and I am not claiming that there is one), it would seem that one could have the weak force particles eat all four of the Higgs bosons, leaving us nothing else to discover. In this regard, it is notable that precision electroweak considerations (see, e.g. here) seem to favor a low value. Based on these indirect considerations, as of July 2011, the data suggest that "The preferred value for its mass, corresponding to the minimum of the curve, is at 92 GeV, with an experimental uncertainty of +34 and -26 GeV," although direct searches have ruled out masses of 114 GeV or less. Thus, the precision electroweak data favor a Higgs mass that is essentially identical to the Z boson mass.
Put another way, suppose that we pretended that the Higgs boson mass was exactly the mass of a Z boson, even though we never observed one (which amounts to assuming that the Z boson has eaten two Higgs bosons), and did all of our Standard Model calculations on that basis. If all of the experimental predictions of this ghost Higgs boson mass value worked to arbitrarily high energy levels, even though no spin zero boson was observed at that mass, there would be no need to discover any more beyond the Standard Model physics. Yet, the current precision electroweak data suggest that we could do just that to energy levels as high as we have ever experimentally observed and to at least some extent beyond that point.
It is also worth noting that the three of the Higgs bosons are "eaten" by weak force bosons in the Standard Model also means that we have already found three of the four Standard Model Higgs bosons, in addition to the vacuum expectation value of the Higgs field (246 GeV), and all of the properties of the Standard Model Higgs boson (not many as a spin zero particle) except its mass and any other properties that are fixed by its mass.
It is also worth observing that in the Standard Model, fundamental particle mass is very intimately tied to the weak force, while being almost entirely decoupled from electromagnetism and the strong force. Hadron mass, in turn, in the Standard Model, arises overwhelmingly from the strong force. (Neutrino mass in the Standard Model, meanwhile, is something of a condundrum.) This fits nicely with the fact that all weak force interacting particles have mass and all particles that don't interact with the weak force lack mass.