Is nature symmetric between particles and antiparticles? Is nature symmetric with respect to presence of particles? Do we have an antiparticle for every particle thought of? Are there any proven examples where we don't have an antiparticle? And what about antiparticle of a photon (we know it can also behave as a particle)? 
 A: Theoretically, for every known particle there is an anti-particle. But bosons are their own antiparticle, for example, the $\pi^+$ which is (mostly) composed of an up-quark and a down-antiquark $u\bar{d}$, has as an anti particle the $\pi^- = d\bar{u}$. The antiparticle of the photon is just the photon, while the antiparticle of the $W^+$ is the $W^-$.
Composite particles, such as the hydrogen atom, also have corresponding antiparticles, but not all of them have been observed. So far the heaviest antimatter atom observed is anti-helium, but theoretically, it seems clear that anti-uranium could be produced.
A: The anti-particle for any particle is obtained by charge C and parity P conjugation. C is the operation that interchanges positive and negative charges and P is the operation that reflects in a mirror. The combined operation of CP must produce a particle of the same mass. This is a theorem of relativistic quantum field theory due to CPT symmetry. This other particle is either the same particle or an antiparticle with opposite charge and/or chirality.
Some particles such a photons, gluons, Z bosons, pions, Higgs, graviton etc, do not have anti-particles because they are invariant under the CP transformation. You can say that they are their own anti-particle. This can only happen for particles without electric charge and with no chirality. 
In principle the QCD color charge is also reversed for an anti-particle. This suggests that a gluon should not be regarded as its own anti-particle, but since colourless states are never seen the distinction cannot really be made in any operational sense.
All known particles which are their own anti-particle are bosons, but it is also possible for a fermion to be its own anti-particle if it is a Majorana spinor rather than a Dirac spinor. No known fermions are of this type (unless neutrinos are Majorana) but they exist in SUSY models.
Observed elementary particles that do have anti-particles include all the quarks and leptons (except possibly the neutrinos) and the charged W bosons. Any composite particle also has an antiparticle made of the anti-particles of its constituents. This can only be its own anti-particle if all its constituents are (e.g. a glueball), or if it is made of particle/anti-particle combinations as is the case for pions.
