Calculating polygon main direction and creating line out of it

A true weighted orientation is non-trivial; I'd say the ST_LongestLine within a polygon is the best approximation, and will yield consistent results over multiple iterations.

• To get your initial Polygons with a column dir in degrees of the main orientation:

  SELECT <some_column_1>,
         ...
         <some_column_n>,
         DEGREES(ST_Azimuth(ST_StartPoint(ll), ST_EndPoint(ll))) AS dir,
         <geom_column>
  FROM (
    SELECT <some_column_1>,
           ...
           <some_column_n>,
           <geom_column>,
           ST_LongestLine(<geom_column>, <geom_column>) AS ll
    FROM <polygon_layer> 
  ) q;
  

• To get the Linestring of the main orientation, with column dir:

  SELECT *,
         DEGREES(ST_Azimuth(ST_StartPoint(geom), ST_EndPoint(geom))) AS dir
  FROM (
    SELECT <some_column_1>,
           ...
           <some_column_n>,
           ST_LongestLine(<geom_column>, <geom_column>) AS geom
    FROM <polygon_layer>
  ) q;