If neither Equator nor the central meridian are aligned with and centered on the map axes, the result is commonly called an oblique projection (or, more properly, an oblique map). Although general properties of the original projection (like area and shape equivalence) still hold, those depending on the graticule orientation are generally not preserved.
Simplified reconstructions of "Oceanic" maps by Spilhaus using Hammer's and August's projections (the original maps have very complicated borders shaped by shorelines, not a round frame). Contrast the same region and aspect in equal-area and conformal projections (the scale in the Hammer map was enlarged 75% so map sizes would be similar)
A common reason for tilting a projection is moving a large, important area to the places of lesser distortion. The Atlantis map (Bartholomew, 1948) presents the Atlantic Ocean in a long, continuous strip aligned with the map's major dimension. Also clearly showing the Arctic “ocean” as a rather small extension of the larger Atlantic, it is an oblique Mollweide projection centered at 30°W, 45°N.
Two other maps emphasizing sea regions were announced by Athelstan Spilhaus in 1942, one using Hammer's and the other August's conformal projection, both with 15°E, 70°S as the map center: very few oceanic sites (notably the Caribbean sea) are interrupted, and relative sizes of oceans are clearly expressed. With modern computers, finding the appropriate rotation parameters for such a “good” distribution of features is fairly easy; one can only imagine the laborious process employed by the original author. Later, Spilhaus also published an ocean-themed world map using a conformal projection in a square and then extended his ideas using interrupted maps.
The pseudoconic equal-area Schjerning IV projection, published in 1904, is an oblique aspect of the well-known Werner projection, with a 0° central meridian but centered on London instead of a pole.
Of course, sometimes a fundamental requirement (like keeping coordinate lines straight or parallel, or preserving correct directions along the meridians in azimuthal projections) prevents adoption of oblique maps.
A fact often overlooked is that points at the borders of any world map are represented at least twice, since in the original sphere the “edges” are joined (an unrelated phenomenon occurs in cylindrical and other flat-polar projections like Eckert's and mine, which stretch the poles into line segments). We are used to that obvious scale distortion (points in a neighborhood get widely separated in the map), so we hardly notice it in conventional maps, save maybe at the extreme tips of Siberia and Alaska.
A similar, very simple modification of Hammer's projection was published by William Briesemeister in 1948: the map is first projected obliquely with 10°E 45°N as the central point; then it is linearly stretched to an aspect ratio of 7 : 4 (compare with 2 : 1 for both Mollweide and conventional Hammer). As a result, parallels in the vicinity of the North Pole are almost circular.
Still another oblique version of Hammer's method is the Nordic projection (Bartholomew 1950), centered at 0°W, 45°N: it closely resembles Briesemeister's map, without the rescaling.