Loyalists and patriots both argued that they sought to uphold the rule of law, how could that be true?

The reason both the loyalists and patriots believed that they were upholding the law was because each group thought that they were doing that. Therfore, the answer to your question rests in their opposing political perspectives.

 Loyalists were labeled as such for the following;

1.  Regardless of geography the British colonies had all the rights of Englishmen

2.  Although colonials, as Englishmen they believed that British law held their interests the same as every other Englishmen. Therefore, the colonies should endure the economic hardships and the politics imposed upon them for the sake of the British Empire just as every other Englishmen would endure

3.  The bare bones reality...many loyalists thought themselves realists...colonial independence from Great Britain was militarily impossible

Patriots were labeled as such for;

1.  Demanding their rights as Englishmen... despite the geography, if taxes were to be increased or new ones added it must be voted upon by those who represent the English people in the Parliament...which included the colonies

2. By the late 1760's early 1770's many political and economic decisions pertaining to the lives of the colonials were being voted upon without their consent

3.  They not only had the right but the duty to rise up against what they saw as a tyrannical government...for the patriot, victory was as just as defeat