What you need to investigate is OmniGraffle's "Choose diagram style" option, available from the process button at the bottom left corner of the OG window.

Diagram Styles can include both the layout direction of the tree and the magnet positions on the nodes.

Switching between the two attached diagram styles will, for example, allow you to change cleanly between top-down and left-right orthogonally linked trees, and new nodes will behave in the manner defined by the prevailing diagram style.

(Just switching layout, rather than diagram style, can cause a mess because the magnet positions will not have changed).

You can create your own diagram styles (File > New Resource > New Diagram Style), and by default, they are stored in:

~/Library/Application Support/OmniGraffle/Diagram Styles/