Computational and model-theoretic properties of logical languages constitute a central field of research in logic-based knowledge representation. Datalog is a very popular formalism, a de-facto standard for expressing and querying knowledge. Diverse results exist regarding the expressivity of Datalog and its extension by input negation (semi-positive Datalog) and/or a linear order (order-invariant Datalog). When classifying the expressiv-ity of logical formalisms by their model-theoretic properties, a very natural and prominent such property is preservation under homomorphisms. This paper solves the remaining open questions needed to arrive at a complete picture regarding the interrelationships between the class of homomorphism-closed queries and the query classes related to the four versions of Datalog. Most notably, we exhibit a query that is both homomorphism-closed and computable in polynomial time but cannot be expressed in order-invariant Datalog.