Am I allowed to make circular references with constants structs?

You are absolutely allowed to do it: add a forward declaration of tail, and C will merge it with a later definition:

typedef struct dlNode {
    const struct dlNode* next, *prev;
    void* datum;
} dlNode;

const static dlNode tail; // <<== C treats this as a forward declaration

const static dlNode head={
    .next=&tail,
    .prev=NULL,
    .datum=NULL
};

const static dlNode tail={ // This becomes the actual definition
    .next=NULL,
    .prev=&head,
    .datum=NULL
};

Note that you should fix your struct declaration to make next and prev constant, otherwise your definition would discard constant qualifiers.

Demo.

You can. You just have to forward declare tail to get it to work:

typedef struct dlNode {
    struct dlNode* next;
    struct dlNode* prev;
    void* datum;
} dlNode;

const static dlNode tail;

const static dlNode head={
    .next = &tail,
    .prev = NULL,
    .datum = NULL
};

const static dlNode tail={
    .next = NULL,
    .prev = &head,
    .datum = NULL
};