Why use a combination of ($E$, $H$) to describe electromagnetic systems instead of ($E$, $B$) or ($D$, $H$)? I attended a lecture on theoretical electrodynamics, and there we exclusively used the combination of E and B to fundamentally describe electromagnetism. They contain no approximation for material behaviour and use the exact charge density and current density (including the bound charges and bound currents in materials).
In the very last lecture we introduced the concept of looking at the system macroscopically, approximating the bound charges and currents in the material as Polarisations P and Magnetizations M and "absorbing them" into the new D and H Field.
D and H thus present an easy way to write the Maxwell equations just for the external charges and currents, where we don't have to precicely know all the the internal ones.
So to my understanding you had these to combinations: You either had to describe the system fully with the combination of (E, B), or you could use the combo (D, H) to factor out bound microscropic charges and currents.
Why then, when I look in many engineering books now, do they use a combination of (E, H) to describe their systems? Why this mishmash of macroscopic framework H and microscopic one E? Why use H where the microscopic processes are factored out, but then also E where they are not?
 A: I was confused by this too when I first encountered it. After all, $\mathbf{E}$ and $\mathbf{B}$ are the "real" fields (in the sense that they directly determine the force experienced by a charge) - so they seem like the natural pair. When I first studied electromagnetic waves in university, everything was in terms of $\mathbf{E}$ and $\mathbf{H}$, with little mention of $\mathbf{D}$ and $\mathbf{B}$. $\mathbf{H}$ also showed up a lot when we learnt about transformers, but in that case $\mathbf{B}$ showed up just as much.
Griffiths gives an explanation along these lines:
The pair $\mathbf{E}$ and $\mathbf{H}$ are commonly seen together because they are easy to measure. In the low-frequency limit:

*

*The line integral of $\mathbf{E}$ gives voltage between the endpoints of the path. Voltages are easily measured with a voltmeter.

*The loop integral of $\mathbf{H}$ gives the (free) current passing through the loop. Free currents are easily measured with an ammeter.

This is only true in the low-frequency limit, since otherwise there are "coupling" terms $\partial\mathbf{B}/\partial t$ and $\partial\mathbf{D}/\partial t$ which will affect the line integrals of $\mathbf{E}$ and the loop integrals of $\mathbf{H}$. Griffiths goes on to say that if it was more natural to measure build-ups of (free) charge than voltage, then we would be seeing $\mathbf{D}$ a lot more often.
Another reason for choosing the pair $(\mathbf{E},\mathbf{H})$ is that it gives a nice form for the Poynting vector, which (in the time domain) is:
$$\mathbf{S}=\mathbf{E}\times\mathbf{H}$$
This corresponds nicely to the formula for power from circuit theory:
$$P=IV$$
Expressing the Poynting vector in terms of other pairs of fields would cause additional constant factors to appear out the front (although technically you could use different units to make these constant factors into 1). Apparently there has been some debate in the past about which form of the Poynting vector should be preferred.
Also note that the usual units of $\mathbf{E}$ are volts per meter (voltage per unit length), and the usual units of $\mathbf{H}$ are amps per meter (current per unit length) - which gives a very nice watts per meter squared for the Poynting vector! So it's as if $\mathbf{E}$ is "voltage-like" and $\mathbf{H}$ is "(free) current-like", which gives nice results for the Poynting vector, for units, and nice relationships to real-world measurements.
Caveat
Griffith warns the reader against thinking of $\mathbf{H}$ as determined only by the free current distribution and somehow "factoring out" the effects of magnetic media. This is a natural conclusion to reach, since in magnetostatics we have $\nabla\times\mathbf{H}=\mathbf{J}_\mathrm{f}$, which is similar to $\nabla\times\mathbf{B}=\mu_0\mathbf{J}$ that gives rise to the Biot-Savart law. However, unlike $\mathbf{B}$, $\mathbf{H}$ does not necessarily have zero divergence, instead we have:
$$\nabla\cdot\mathbf{H}=-\nabla\cdot\mathbf{M}$$
Which means that $\mathbf{H}$ also depends on the magnetization of the medium. However, loop integrals of $\mathbf{H}$ will always be equal to the free current passing through the loop (in magnetostatics at least, since otherwise the displacement current term $\partial\mathbf{D}/\partial t$ will be non-negligible).
A: Physics is littered with various notational conventions. Generally people will use whatever combination of variables and whatever system of units they find most convenient at the time for their purpose. In electrodynamics one would generally use whatever combination of variables which makes Maxwell's equations "look the simplest" given the presence of sources, bound currents/charges, etc. I know of no reason other than pure convenience.
Any formulation of Maxwell's equations in terms of $(E,H)$ or $(E,B)$ or $(D,H)$ is formally equivalent.
A: If my memory is correct, this is because they are the easiest (or most common) to measure.
A: History
There is a long history of using fields $\mathbf{E}$ and $\mathbf{H}$ together, which is reflected in their names and notations: $\mathbf{E}$ and $\mathbf{H}$ are sometimes called tensions, while $\mathbf{D}$ and $\mathbf{B}$ are called displacements/inductions (I am not sure how generally accepted these terms are in English, but other languages do make this distinction). This is further reflected, e.g., in the form of the constitutive relations:
$$
\mathbf{D}=\varepsilon\mathbf{E}, \mathbf{B}=\mu\mathbf{H}
$$
(rather than $\mathbf{H}=\mu\mathbf{B}$).
Macroscopic electrodynamics
When dealing with macroscopic Maxwell equations, any pair of electric and magnetic fields can be used without causing ambiguity:
$$
\mathbf{E}\text{ and }\mathbf{H}\\
\mathbf{E}\text{ and }\mathbf{B}\\
\mathbf{D}\text{ and }\mathbf{H}\\
\mathbf{D}\text{ and }\mathbf{B}
$$
Miscroscopic electrodynamics
In microscopic electrodynamics the distinction between the tensions and the induced fields disappears. In Gaussian units they are simply equal:
$$
\mathbf{D}=\mathbf{E},\mathbf{B}=\mathbf{H},
$$
so again the use of specific letters is a matter of convention in a particular field.
