ST_Overlaps — Tests if two geometries have the same dimension and intersect, but each has at least one point not in the other
boolean ST_Overlaps(
geometry A, geometry B)
;
Returns TRUE if geometry A and B "spatially overlap". Two geometries overlap if they have the same dimension, their interiors intersect in that dimension. and each has at least one point inside the other (or equivalently, neither one covers the other). The overlaps relation is symmetric and irreflexive.
In mathematical terms: ST_Overlaps(A, B) ⇔ ( dim(A) = dim(B) = dim( Int(A) ⋂ Int(B) )) ∧ (A ⋂ B ≠ A) ∧ (A ⋂ B ≠ B)
This function automatically includes a bounding box comparison
that makes use of any spatial indexes that are available on the geometries.
To avoid index use, use the function 
Performed by the GEOS module
Enhanced: 3.0.0 enabled support for 
NOTE: this is the "allowable" version that returns a boolean, not an integer.
This method implements the OGC Simple Features Implementation Specification for SQL 1.1. s2.1.1.2 // s2.1.13.3
This method implements the SQL/MM specification. SQLMM 3: 5.1.32
ST_Overlaps
returns TRUE
in the following situations:



A Point on a LineString is contained, but since it has lower dimension it does not overlap or cross.
SELECT ST_Overlaps(a,b) AS overlaps, ST_Crosses(a,b) AS crosses, ST_Intersects(a, b) AS intersects, ST_Contains(b,a) AS b_contains_a FROM (SELECT ST_GeomFromText('POINT (100 100)') As a, ST_GeomFromText('LINESTRING (30 50, 40 160, 160 40, 180 160)') AS b) AS t overlaps  crosses  intersects  b_contains_a ++ f  f  t  t
A LineString that partly covers a Polygon intersects and crosses, but does not overlap since it has different dimension.
SELECT ST_Overlaps(a,b) AS overlaps, ST_Crosses(a,b) AS crosses, ST_Intersects(a, b) AS intersects, ST_Contains(a,b) AS contains FROM (SELECT ST_GeomFromText('POLYGON ((40 170, 90 30, 180 100, 40 170))') AS a, ST_GeomFromText('LINESTRING(10 10, 190 190)') AS b) AS t; overlap  crosses  intersects  contains +++ f  t  t  f
Two Polygons that intersect but with neither contained by the other overlap, but do not cross because their intersection has the same dimension.
SELECT ST_Overlaps(a,b) AS overlaps, ST_Crosses(a,b) AS crosses, ST_Intersects(a, b) AS intersects, ST_Contains(b, a) AS b_contains_a, ST_Dimension(a) AS dim_a, ST_Dimension(b) AS dim_b, ST_Dimension(ST_Intersection(a,b)) AS dim_int FROM (SELECT ST_GeomFromText('POLYGON ((40 170, 90 30, 180 100, 40 170))') AS a, ST_GeomFromText('POLYGON ((110 180, 20 60, 130 90, 110 180))') AS b) As t; overlaps  crosses  intersects  b_contains_a  dim_a  dim_b  dim_int ++++++ t  f  t  f  2  2  2