What is the magnetic counterpart of $E$ in electromagnetism, $H$ or $B$? The name, magnetic field, denoted $H$, suggests that it is the counterpart of the electric field $E$. And therefore $B$ must the one for $D$. But I also find (what I judge as good) reasons for having $B$ as the natural counterpart of $E$. In the following I'll denote "$\sim$" as "is the physical counterpart of".
Reasons for having $B\sim E$:


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*The Lorentz force is $F=q(E+v\times B)$, which suggests that $E$ and $B$ are "natural counterparts" of each other.

*The induction law (of the Maxwell equations) naturally relates $E$ with $B$ without further constants.

*A distribution of charges produces $D$, which does not change when a dielectric is introduced (but $E$ changes). Similarly, a distribution of currents creates H, which is independent of the material in which these charges is embedded (while $B$ changes). Therefore $E$ and $B$ behave similarly, i.e. they are sensitive to the presence of materials.

*For dielectric materials, we have $D=\epsilon_0 E+P$ while for diamagnetic materials, we have $B=\mu_0(H+M)$ (note the parenthesis). This implies that $D$ has the same units a $P$, polarisation. For the same reasons, $H$ has the same units as the magnetisation $M$. Therefore $D\sim H$, not $B$
Reasons for having $H\sim E$:


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*The poynting vector is $S=E\times H$ without further constants

*If we had defined $M$ such that $B=\mu_0H+M$ (do you see a reason why this is not possible?), then the two expression for the fields within materials would be equivalent and we would have $B\sim D$.
The arguments 1 and 2 for $B\sim E$, as well as 1 for $B\sim E$ are rather weak, I know. We need constants to make stuff dimensionally correct, and having a  cleaner equation is not a good physical reason. Argument 4 for $B\sim E$ and 2 for $H\sim E$ are the same, and I guess that it has historical roots. But I'm pretty sure you can go for both options and everything would work fine.
But I find argument 3 for $B\sim E$ especially physical and intuitive. Actually, the counter-intuitive behaviour of $H$ vs. $E$ on their dependence on the material has always puzzle me and led to post this question.
What do you think is the physical counterpart of the electric field? Why? Is all this a silly misunderstanding motivated by historical developments/naming conventions? Or there profound reasons for this counter-intuitive mix of behaviours? 
 A: In most of the literature the symbols 'E' and 'B' are reserved for applied fields. So the equivalence depends on the context. The applied fields E and B creates a response in electric or magnetic materials and the effect of them are given by D (displacement field) or H-field. So in the vacuum (free case), E and B are the relevant equivalent fields and whenever there is matter present D and H are relevant fields. So the free Maxwell equations contain E and B and within matter it contains D and H. In most cases, the magnetic materials are considered only magnetic (no electric effect) and it is easier to deal with, we use the definition $\vec{S} = \vec{E}\times \vec{H}$ (since it follows directly from Poynting theorem). But this does not in any way confirm that H is the equivalent of E, it is just dimensionally consistent. I quote here from Wikipedia

It is also possible to combine the electric displacement field D with the magnetic field B to get the Minkowski form of the Poynting vector, or use D and H to construct yet another version. The choice has been controversial: Pfeifer et al. summarize and to a certain extent resolve the century-long dispute between proponents of the Abraham and Minkowski forms (see Abraham–Minkowski controversy).

A more trustworthy source to check the equivalence of E and B is the tensor formalism (covariant formalism) of Electromagnetism. There if you check the stress energy tensor for electromagnetism, E and B appear together. In more advanced mathematical language E and B are Hodge dual (kind of like dual space but not the same) of each other. So E and B are the more fundamental quantities which does not depend upon the response of a material and are equivalent.   
