To return to the analogy of the Necker Cube, the mental flip that I want to encourage can be characterized as follows. We look at life and begin by seeing a collection of interacting individual organisms. We know that they contain smaller units, and we know that they are, in turn, parts of larger composite units, but we fix our gaze on the whole organisms. Then suddenly the image flips. We see through them to the replicating fragments of DNA within, and we see the wider world as an arena in which these genetic fragments play out their tournaments of manipulative skill.
Genes manipulate the world and shape it to assist their replication.
It happens that they have 'chosen' to do so largely by moulding matter into large multicellular chunks which we call organisms, but this might not have been so. Fundamentally, what is going on is that replicating molecules ensure their survival by means of phenotypic effects on the world. It is only incidentally true that those phenotypic effects happen to be packaged up into units called individual organisms.