I reproduced your problem and noticed that rotating behavior depends on the marker types used to create your multi-layer marker:
Use character markers instead of other marker types if you want a multi-layer symbol to be rotated as a whole.
Simple Marker, Arrow Marker and Picture Marker layers rotate individually.
Each Simple Marker, Arrow Marker and Picture Marker layer rotates about its own center. This eventually breaks the layout of your multi-layer symbol if non-zero offsets are used.
(Above example: rotation of a symbol, consisting of multiple simple markers and an arrow marker.)
Character Marker Layers always rotate about the common origin (0,0)
Use character marker symbols for all layers with non-zero offset in order to avoid individual rotation of your symbol layers. However, you are bound to font symbols and their available characters. (If you are looking for simple arrows and such, most probably you find what you need.)
(Above example: rotation of a symbol, consisting of multiple character markers.)
Avoid rotation and placement issues by using a single layer symbol
If these options do not fit your needs, you could design your marker symbol with any graphics software (e.g. Inkscape or Illustrator for vector graphics) and import it to ArcGIS as a single picture marker symbol. (Possible file formats include emf (vector graphics) or png or bmp (raster graphics).)
The "silver bullet" would be to create your own font (this also supports anti-aliasing in contrast to picture marker symbols) and use a single character marker symbol. However, this is most likely disproportionate (and inflexible to changes) for a single symbol.