Maxim treats the cocker spaniel Jasper as an ordinary family pet, generally showing him some affection but not actually giving him much thought. As the novel wears on, Jasper seems to become more of a companion to the narrator rather than Maxim; he is often described as following her about the house and going out with her. The one time that both the de Winters take him for a walk, Maxim ends up becoming rather impatient with him, calling him a ‘damn dog’ (chapter 10); although it is true that at this point Maxim is irritated with him for running towards Rebecca’s old cottage on the beach, which holds some very unsavoury memories for Maxim.
The real significance of the relationship between Maxim and Jasper is that the narrator views it as symbolic of the way that Maxim treats her. This is made explicit quite early on in the novel, when Beatrice and Giles first come to visit, and they all sit outside together. The narrator sits close to Maxim while he and Beatrice talk:
I listened to them both, leaning against Maxim’s arm, rubbing my chin on his sleeve. He stroked my hand absently, not thinking, talking to Beatrice.
‘That’s what I do to Jasper,’ I thought. ‘I’m being like Jasper now, leaning against him. He pats me now and again, when he remembers, and I’m pleased, I get closer to him for a moment. He likes me in the way I like Jasper.’ (chapter 9)
Here, Maxim hardly appears aware of the narrator’s presence at all, apart from to automatically pet her as she snuggles up to him; he does not talk to her, only to his sister Beatrice.
Generally speaking, Maxim does show the narrator affection, while also getting annoyed with her at times (as with Jasper), but she resents the fact that he doesn’t seem to treat her as an equal, or indeed, as an intellectually-functioning human being at all. Later on, after the fiasco of the ball, she again brings up this image, reflecting miserably how she loves Maxim ‘in a sick, hurt, desperate kind of way, like a child or a dog’ (chapter 18).
It is not until after the ball, with the unexpected discovery of Rebecca’s boat and body, that things change between the de Winters. The narrator realizes that Maxim remained distant from her only because he was so troubled by his hellish memories of Rebecca, not because he regarded her as inferior. Now, finally, he opens up to her, revealing everything. Finally, they become equals, and he comes to rely heavily on her love and support, just as she relies on him.