If you look at the trend in orbital energies as you go across a period the pattern is clear (orbital energy decreases with increasing effective nuclear charge) and, to my knowledge, it has no anomalies like the trend in ionization energies does when a 2p orbital is filled for the first time with two electrons (when this happens the ionization energy decreases). I expected that the graphs for ionization energy would be mirror images in the x-axis given that orbital energies are negative and measured relative to an ionized electron (which equals 0) albeit with different y-scales because ionization energy is the energy to ionize a mole of a substance in a gaseous state. Is my understanding that orbital energy = -(energy required to remove one electron from that shell) incorrect? Fundamentally, the question is, why doesn't the trend in orbital energies mirror the trend in ionization energies?
First off, the idea that the orbital energy is the negative of the ionization energy is actually Koopmans' theorem. It works qualitatively because of cancelation of errors.
But I digress...
Orbitals are single-electron constructions. That is, in Hartree-Fock theory, an orbital is a single-electron wavefunction assuming the average electrostatic potential of the remaining electrons.
Ionization energy trends are experimental observables and you see effects of multi-electron interactions.
So the short answer is that orbital energies are not physical quantities (i.e., being single-electron mathematical results) and ionization energies are based on multi-electron interactions.
Probably the simplest way to understand the trends in ionization energies vs. orbital energies is to look at Slater's rules and also remember Hund's rules. That is, when we add electrons, we are partially shielding the nuclear charge, but we also have to consider the stabilization due to exchange of electrons and destabilization when we pair electrons in an orbital. These are multi-electron effects, and only some of that can be recovered in a simple one-electron picture.
For the $2p$ orbitals, as you mention, there's a decrease in the ionization potential when you put two electrons in the same orbital. This happens because this new electron does not benefit from the exchange effect (Hund's rules) and has an electrostatic repulsion with the electron in the same orbital. Thus it's easier to remove it than might be expected from a one-electron orbital picture.