Paper:
Representations learnt through deep neural networks tend to be highly informative, but opaque in terms of what information they learn to encode. We introduce an approach to probabilistic modelling that learns to represent data with two separate deep representations: an invariant representation that encodes the information of the class from which the data belongs, and an equivariant representation that encodes the symmetry transformation defining the particular data point within the class manifold (equivariant in the sense that the representation varies naturally with symmetry transformations). This approach to representation learning is conceptually transparent, easy to implement, and in-principle generally applicable to any data comprised of discrete classes of continuous distributions (e.g. objects in images, topics in language, individuals in behavioural data). We demonstrate qualitatively compelling representation learning and competitive quantitative performance, in both supervised and semi-supervised settings, versus comparable modelling approaches in the literature with little fine tuning.

Review:
This paper is well written, and the quality of the figures is good. In this paper, the authors propose an invariant-covariant idea, which should be dated back at least to the bilinear models. The general direction is important and should be pursued further. 
However, the literature is not well addressed. Eslami et al. 2018 have been cited, but some very important and related earlier works like: 
[1] Kulkarni et al. 2015, Deep Convolutional Inverse Graphics Network
[2] Cheung et al. 2015, Discovering Hidden Factors of Variation in Deep Networks
were not discussed at all. The authors should certainly make an effort to discuss the connections and new developments beyond these works. At the end of section 1, the authors argue that the covariant vector could be more general, but in fact, these earlier works can achieve further equivalence, which is much stronger than the proposed covariance.
There is also an effort to compare this work to Sabour et al. 2017 and the general capsule idea. I would like to point out, the capsule concept is a much more fine-grained what & where separation rather than a coarse-grained class & pose separation in one shot. In a hierarchical representation, what & where can appear at any level as one class can consist of several parts each with a geometrical configuration space. So the comparison of this work to the generic capsule network is only superficial if the authors can not make the proposed architecture into a hierarchical separation. Besides different capsule network papers, I found another potentially useful reference on a fine-grained separation:
[3]Goroshin et al., Learning to Linearize Under Uncertainty
In the paper, it is argued several times that the latent vector r_y contains a rich set of global properties of class y, rather than just its label and the aim is that it can learn what the elements of the class manifold have in common. But this point is not supported well since we can always make a label and this latent vector r_y equivalent by a template. I think this point could be meaningful if we look at r_y's for different y, where each of the dimension may have some semantic meaning. Additional interpretation is certainly needed.
Under equation (3), "Note that v is inferred from r_y" should be "inferred from both r_y and x", which is pretty clear from the fig 5. Related to this, I could imagine some encoder can extract the 'style' directly from x, but here both r_y and x are used. I couldn't find any guarantee that v only contains the 'style' information based on the architecture with even this additional complication, could the authors comment on this?
Equation (5) is not really a marginalization and further equation (6) may not be a lower bound anymore. This is probably a relatively minor thing and a little extra care is probably enough.
The numbers in table 2 seems a little outdated.
To conclude, I like the general direction of separating the identity and configurations. The natural signals have hierarchical structures and the class manifold concept is not general enough to describe the regularities and provide a transparent representation. Rather, it's a good starting point. If the authors could carefully address the related prior works and help us understand the unique and original contributions of this work, this paper could be considered for publication.