What are parahelium and orthohelium? I have been learning a little about two-electron atoms, and there are some things that I do not fully comprehend. Some context:
In the two books I have been reading (Physics of atoms and molecules, by Bransden & Joachain, and Quantum Mechanics of one- and two-electron atoms, by Bethe and Salpeter), ortho and para eigenstates are mentioned when discussing the Pauli principle. Since the global wavefunction of the two-electron atom needs to be anti-symmetric due to the exclusion principle, eigenstates of the Hamiltonian that are symmetric (para states) can only have a spin wavefunction from the spin singlet (anti-symmetric). Anti-symmetric eigenstates (ortho states) can only haven an associated spin wavefunction from the spin triplet (symmetric). This makes perfect sense to me in an abstract way, but then two things are mentioned: ortho helium and para helium, two types of helium, each with its own energy level diagram. Parahelium is different because the lower energy state allowed is the 1s orbital, which has an associated orbital harmonic function which is symmetric. Each energy level (defined by two quantum numbers, $n$ and $\ell$) can be inhabited by a single electron. Orthohelium does not have this energy level at all, and each energy level can be inhabited (if I understood correctly) by three electrons, because $M = 2S + 1 = 2\times 1 +1 = 3$. This is what I gathered from reading the two relevant sections in the aforementioned books.
My question is: what are these different helium varieties? I would imagine, even if there are two possibilities (orthohelium with spin triplet electrons and parahelium with spin singlet electrons), real helium gas would be comprised of a mix of them. Therefore, measuring its spectrum would yield a mix of the two different spectra. How could early 20th century spectroscopists tell they where two superimposed spectra, instead of considering the mix of lines one single spectrum?
I think I'm failing to see the connection between theory and experiment.
 A: The reason why it seems like there's two different types is because it's very difficult to drive spin-flip transitions, which are what is required to switch from a singlet state to a triplet state. The lifetime of the metastable triplet ground state is around 8000 seconds, which is an eternity in atomic physics terms, and an indication of how hard it would be to drive the transition with electromagnetic radiation of the corresponding wavelength. Typical lifetimes are usually somewhere on the order of nanoseconds (though there is a wide range).
The most common way for atoms to transfer from one type to the other is through collisions with electrons, where the restrictions on angular momentum conservation (which is ultimately what makes these transitions so hard to drive) are much easier to satisfy. It also requires 3.2 eV of energy to drive it, and so collisions at room temperature (~1/40 eV) will not produce any significant amount of triplet helium.
A common way to look at atomic spectra (at least back in the days before lasers) was to excite the atom in a discharge cell, which essentially works like a fluorescent lamp and produces a plasma. Collisions between electrons and atoms in the plasma will excite atoms into pretty much every available state, and so all helium lines corresponding to allowed transitions will be visible as emission lines.
Typically with similar electronic structures the singlet state is the lower energy state, but molecular oxygen is one well-known instance where the ground state is a triplet state, which has a variety of interesting impacts on its properties from magnetism to reactivity. The energy difference here is about 1 eV so again room-temperature collisions will not produce more than a minuscule amount of singlet oxygen.
A: Early 20th century spectroscopists measured the helium spectrum with sufficient accuracy to clearly establish that helium had two sets of spectral lines that seemed incompatible with each other. There was no single set of energy levels that could accommodate the observed lines without predicting other lines that were not observed. The phenomenological solution was to postulate the existence of two types of helium, ortho- and para-, each with their own energy levels.
In particular, Bohr's quantum mechanics could not provide an explanation, and this was a major motivation for the development of the new quantum mechanics in the 1920's.
Quoting from Birthwistle's 1928 "The New Quantum Mechanics", Chapter XXVI:

It is well known that the spectral terms of helium can be divided into two sets such that no term of the one will combine with a term of the other to produce a spectral line.
… One set by its transitions gives the 'para helium' lines … the other set gives the 'ortho helium' lines …
The obvious failure of the classical mechanics and the correspondence principle to solve the problem of a nucleus with two outer electrons was one of the factors which compelled Heisenberg to seek for a new quantum mechanics …

So the connection between experiment was very tight. Early precision experiments could not be explained by existing models, so theorists worked very hard to develop a new (and ultimately hugely successful) theory.
I have asked a question over on History of Science and Mathematics to see if anyone there knows more details on the early history of the helium spectrum measurements and the challenge to early quantum mechanics. I will update this answer if I find out more.
