"in order that" vs "in order to"

Both formulations are valid. But IMO the former in order to makes the desired outcome the direct result of the action, while the latter in order that suggests the outcome could be a higher-order consequence e.g. of a causal mechanism that is implied but not specified.

In order may be followed either by an infinitive clause,
which is introduced by the infinitive complementizer (for...)to,
or by a tensed clause, which is introduced by the complementizer that.

• He left early in order for Mary to make the presentation.
• He left early in order to make the presentation.
• He left early in order that there would be enough light to drive.
• He left early in order that we would be free to follow him.
• He left early in order that we be free to follow him.

The last example is a fairly rare construction
(called "present subjunctive" by some)
that implies possibility but not necessity.

The first example (in order to) seems more formal in the way the idea is expressed. On the other hand, the second example (in order that) has an insistence on the word "we", and makes it feel as the author has an emotional attachment to the subject. They both mean the same thing, but the emotion they translate to the reader, the first one having little, the second one having an inclusive subject, is the nuance.

"In order to" feels a lot more general in its approach. "In order that" insists on the subject that follows.