What is the relationship between assertion and reference?

See Speech Acts:

In the philosophy of language and linguistics, a speech act is an utterance considered as an instance of action in a social context rather than as the mere expression of a proposition.

Types of speech act: following J. L. Austin's distinction in How to Do Things with Words (1955), many philosophers and linguists analyse ordinary utterances as involving at least three kinds of act: a locutionary act (producing a meaningful linguistic expression), an illocutionary act (performing an action such as asserting, questioning, or promising in saying that expression), and often a perlocutionary act (bringing about further effects on an addressee, such as convincing or alarming them).

A grammatical sentence composed of meaningful words is commonly thought to express a “content,” which is determined by what that sentence literally means together with features of the context of utterance. [...] For indicative sentences, such contents are typically called propositions. Propositions, then, are the contents of indicative sentences, are what such sentences express, and, further, are often thought to be the primary bearers of truth value.

Merely expressing the Proposition that it is snowing is not to make a move in a “language game”. Rather, such a move is only made by putting forth a Proposition with an illocutionary force such as assertion, conjecture, command, etc. [...] The force/content distinction also finds parallels in our understanding of mentality. Speech acts are not only moves in a “language game.” They also often purport to express of states of mind with analogous structural properties. An assertion that it is snowing purports to express the speaker’s belief that it is snowing.

Reference is a relation that obtains between a variety of representational tokens and objects or properties: typically, a name and the individual named.

In conclusion, we have two basic lingustic constructs: terms (names) that we use to refer to something and sentences (propositions) that we use to assert something.

But we can conflate the two. See Frege's assertion sign:

The sign ⊦ [also called judgement stroke] was used by Frege to put in front of sentences that are being asserted as true, as opposed to sentences that are involved in some process of reasoning, but are not themselves put forward as true.

The key-point is that for Frege (but modern logicians and philosophers do not agree) also sentence are names: they are name of truth-values.

Useful resource: Mark Jary, Assertion (2010, Palgrave)

OP: "Psychologically, it seems that assertion and reference are both indispensable. When I refer to some object, I hopefully am committed to there being such an object. …
Nevertheless, it seems impossible to do without both. We cannot assert, without having something to refer to prior …"

When the Higgs boson was first posited as a possibility in 1964 no-one had ever seen one and it only existed as a theoretical object. Prior to discovery in 2012 its concept was even ambiguously constructed: its most likely mass region was predicted as 115–130 GeV but it could also have been 130–180 GeV, which goes to show that an assertion can be vague. Upon discovery the Higgs boson was found at about 125 GeV. Obviously, not all theoretical objects turn out to be actual objects. Prior to discovery it could not have been positively asserted to have existed in actuality; after discovery it can be confidently asserted to have existed prior to discovery.

There is a considerable quantity of philosophical literature on the subject of reference. If you are interested in exploring it, you might start with the SEP article on the subject.

Reference is usually understood to be concerned with the relationship between signs, symbols and linguistic tokens that have some kind of representational function and the objects or concepts that they refer to. Many things have a reference, such as names, definite descriptions and indexical expressions. There are several theories about how they come to refer, including descriptivist and causal theories. Some philosophers, such as Paul Grice and Peter Strawson, claim that it is agents (people) who refer to things, rather than the signs and tokens themselves, and so they prefer to think of reference in terms of a speaker's intentions.

Things do not have to exist in the real world for us to refer to them. The name Luke Skywalker refers to a fictional individual. The word phlogiston refers to a concept that has been discarded by scientists as having no explanatory or predictive value. We can refer to non-existent things and make assertions about them, e.g. "Santa Claus does not exist". We can refer to things that have potential or possible existence, e.g. "We should preserve the planet for future generations". We can refer to abstract objects such as numbers or the concept of beauty. We can refer to things whose existence is unknown or disputed.

Assertion is a kind of speech act. It is an action performed by an agent over and above merely referring to, mentioning or conjecturing a proposition. To make an assertion is to make a claim or judgement that something is the case. We can refer to the proposition that the Eiffel Tower is tall without asserting it. For example, I might say, "John thinks that the Eiffel Tower is tall", "Suppose for the sake of argument that we all agree that the Eiffel Tower is tall", "If the Eiffel tower is tall then so is the Empire State Building", "It is not the case that the Eiffel Tower is tall". The last of those may be used as a speech act of denial of the proposition that the Eiffel Tower is tall.

You ask:

Is there a deep reason, perhaps, why we need these two distinct “psychological modes”, and could never do away with either of them?

Yes, both are tools in the Wittgensteinian language game. With reference, we are able to bring about associations and imply things, and with assertion, we are able to make claims about things, relations, and properties. Therefore, assertions allows us to explicitly communicate and discuss logical consequence. Both of these are considered forms of explicature, and explicature helps us increase the information created to determine relevance to decide on meaning, at least according to relevance theory. Both also can be used within broader contexts to communicate by implicature if speakers prefer to efface meaning for reasons of secrecy, politeness, and so on. As for their relation, reference is prelinguistic and can be used by primates and very young children, but as the human faculty for language develops, grammar allows the construction of utterances that encode assertion. Assertion presupposes reference.

You also say:

[Y]et every act of referring is implicitly a kind of assertion.

Yes, this is often referred to as ontological commitment by way of Quine. When we refer to things, at the outset, we often presume they exist before proving them. Thus, a claim "water is H20" seems to presume the existence of both water and H20. For more information on that, see "Presuppositions" (SEP). Quine also went to say, following Carnap's notion of formal mode, that negotiating and communicating those ontological commitments is called semantic ascent. Ultimately for Quine, existential quantification (SEP) was the threshold of whether or not something could be committed to.

There is obviously a "deep" reason for this, but a complete exposé of it would require an expertise about logic no one currently has.

In broad terms, an assertion is necessarily an assertion about something, even something which is not specified in any way (except that it be something). Consequently, one way or the other we need to specify the thing we mean to make an assertion about, and then we need to make the assertion itself: Subject, predicate.

Part of the reason for this is our epistemological position as observer. It cannot be reasonably denied that we think about the world in terms of things and their properties, although it would be less misleading to say that we think of the world in terms of classes of things. As Immanuel Kant could have said, we do not know the thing in itself, so all we can do is believe that it belongs to this or that class of phenomena. The assertion that A is B is essentially an assertion that if a thing is A, then it is B, which is to say that the class of things which are A is included in the class of things which are B.