Under confinement, the average shape of a polymer chain is modified in interesting ways. In this work, we discuss how confinement affects the mean geometrical properties of wormlike polymers with variable flexibility and monomer-monomer interaction. Here, we consider a polymer mushroom, i.e., a single chain that is permanently anchored to a flat surface by an end point. Compression is introduced by confining the chains inside an infinite slab with parallel hard walls. Regarding polymer shape, we focus on two large-scale geometrical properties that are not correlated a priori: the chain's size and its entanglement complexity. Using Monte Carlo simulations, we have analyzed the behavior of these two properties under confinement for a range of potential energy functions. A recurrent pattern of shape transitions emerges, as indicated by changes in the correlation between mean size and entanglements. Our results show that, whereas a flexible polymer with strong self-attraction sustains high compression without deforming, polymers that are either too rigid or too weakly self-attracting are "flattened" by slight compression. Furthermore, we find a general relation between molecular size and entanglements that is valid over a range of polymer models and levels of confinement. We conclude that chain stiffness influences less the compressive behavior of a polymer than chain self-interactions.