He worked hard to protect his family's honor.
会社の体面を保つために、問題は公にされなかった。
To maintain the company's reputation, the issue was not made public.
体面 is a more formal, written-register word for social standing, while メンツ is the colloquial term used in everyday speech about not losing face.
体面 is a more formal word for public dignity or appearances, while 面目 focuses on one's personal honor or reputation among others.
体面 (same reading as 対面) means one's reputation or public image, not a face-to-face meeting like 対面.
Compound of 体 (body, appearance) and 面 (face, surface), literally 'face and appearance'. The meaning of social standing or dignity is an extension of the idea of one's outward face shown to society.