How are these to be reconciled with this hymn?
Very simply.

The rite is described and verses from the hymn are injected into it without the slightest logical connection.

That is the essence of all the Brahmanic ritualism.

The later rite is as follows: Three altars are erected, northwest, southwest, and southeast of a mound of earth.

In the fourth corner is the corpse; at whose feet, the widow.