Is there a way to reorder the sections of the page templates in the dialog that is opened when creating a new page or changing the page template of an existing page?
How to reorder the page templates section in the page templates' dialog?

Hide unwanted templates while creating page

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In new interface:
There is no option to change the order of the sections in the page templates' dialog.

In classic interface:
The order of the sections is based on the Ordinal property of the page template categories. Execute the following query in order to compare the currently set order of the properties:

SELECT ordinal, *
FROM sf_taxa
WHERE taxonomy_id IN (SELECT id
  FROM sf_taxonomies
  WHERE nme = 'PageTemplates') 
AND id <> 'F669D9A7-009D-4D83-AABB-000000000003'

In order to change the Ordinal property, refer to the following sample for a reference:

TaxonomyManager manager = TaxonomyManager.GetManager();

HierarchicalTaxonomy pageTemplatesTaxonomy = manager.GetTaxonomies<HierarchicalTaxonomy>().Where(t => t.Name == "PageTemplates").FirstOrDefault();

if (pageTemplatesTaxonomy != null)
{
    IEnumerable<Taxon> pageTemplatesTaxa = pageTemplatesTaxonomy.Taxa.Where(t => t.Id != Guid.Parse("F669D9A7-009D-4D83-AABB-000000000003"));

    foreach (var taxon in pageTemplatesTaxa)
    {
        // apply conditions and set the Ordinal property to the desired value based on the conditions
        // taxon.Ordinal = 4;
    }

    manager.SaveChanges();
}