Asymmetries and shifts of the hysteresis loop give information about material properties which need to be discussed in context of structure, anisotropies,... system set-up and experiment.

The area enclosed in a hysteresis cycle (plotted in proper units) represents the irreversible work required to go through the cycle. Often enough, a single hysteresis cycle cannot reveal much physics beyond such qualitative quantities including squareness etc., which you mention. However, in many cases this is exactly the practically relevant information and thus may prove sufficient.

If you can come up with a specific model of magnetization reversal in your specimen or system then you may be able to check the compatibility of the experimental dataset with your model and determine the parameters of that particular model from the shape of the hysteresis curve. Very different physical mechanisms of magnetization reversal may produce very similar shaped M(H) loops, though.

In case you can produce a dataset that not only includes a so-called "major loop" but also "minor loops", then there exist intelligent recipes to conduct such experiments in a way as to reveal quite a bit about the nature of magnetization reversal in a specimen.

Since finite hysteresis is a sign of non equilibrium, chosing (if available) to perform hysteresis cycles with an appropriate set of field sweep rates may reveal some information about the dynamics in your system.

Furtheron you may look for temperature dependencies, directional anisotropies etc., all of which do relate to specific properties which may be of interest.

Also, you can obtain cualitative information about magnetic exchange interactions between the nanoparticles in the sample. It is very useful to study spring magnets.

I totally agree with all comments given above. In addition, I would remark that is very important to know the nature of your material (If it is a polycristal, is a set of nanoparticles, is anysotropic... ) to interpret the M(H) loops observed.

Additional M-T measurements as ZFC-FC curves and M(H) curves at different temperatures and/or magnetic relaxation measurements would be of special interest to model, test and properly interpret the magnetic properties of your material.

I also agree with all comments. Furthermore, it is important that to know your materials type I mean is it paramagnetic, ferromagnetic or ferrimagnetic..etc. Each type has own features when you want to study the hysteresis loop (shape of the loop). Also, it is interesting to study the magnetization with the change the direction of magnetic field (in-plane or out of plane). Also, measuring the initial magnetization is also interesting.

From the M-H hysteresis curve, we can confirm the type of the materials, ferromagnetic or anti-ferromagnetic, or parametric. In addition is this is soft ferromagnetic or hard ferromagnetic???. Moreover, We can experimentally detect the spin state of the sample, LS, IS, and/or HS states in a simple way. If we study M-H with changing temperature we can also detect the transition temperature (TC) and the phase separation and phase boundary too. Moreover, we can detect the swishing shape of the curves.

Considering the Temperature evolution of the M(H) curves and in particular of the coercive field, it is also possible to distinguish between the pinning mechanisms (weak or strong) that occur in your systems. This study is very useful not only for bulk samples but also for thin films with hard magnetic character, for example, or also in exchange spring films (i. E. hard -soft coupled bilayers)

One can also try approach-to-saturation law on virgin curve at high fields to get some insights into the nature of magnetic anisotropy, high field susceptibility, etc.,. The information about this can be found in any basic magnetism book.

I also agree with all given comments.

Just add a pieces, first of all: what kind of specimen u have, how u treat it, what is the instrument (capability), how u prepare the sample for analysis etc.

Example: u can plot a minor loop (isothermal magnetization-demagnetization) to know how the exchange interaction work in your specimen.

M-H curve is sensitive with the external filed direction relative to the sample anisotropy (crystal and or shape). U can get a lot more info, even u can tailoring ur specimen by understand this curve only.

There a few model to fit the curve that u can use to extract some more information such as Energetic model, Hodgdon, Langevin, Jiles-Atherton, Preisach etc.

Hope it help.

By differentiating lower branch of the hysteresis curve you obtain the so called switching field distribution.