Th standard idiom is "have [one's] head screwed on right [or "on the right way"]." Both British English and U.S. English have this idiom, but the sense of the phrase may differ slightly. The sense in British English seems to be essentially what FumbleFingers reports in a comment above. John Ayto, Oxford Dictionary of English Idioms, third edition (2009) has this entry for the phrase:
have your head screwed on (the right way) have common sense. informal
And Cambridge International Dictionary of Idioms (1998) has this:
have your head screwed on (the right way) informal
if someone has their head screwed on the right way, they do not do stupid things [example omitted]
But in U.S. usage, the phrase may imply less of an opposition between stupidity and common sense, and more of one between erratic behavior/eccentricity and reliability/predictability. Here is the entry for the idiom "have a screw loose" in Christine Ammer, American Heritage Dictionary of Idioms (1997):
have a screw loose Be mentally unstable or eccentric, as in Anyone who approves that purchase must have a screw loose. This term likens a mental weakness to a machine in which a part is not securely fastened. An antonym is have one's head screwed on right; for example, She's very capable; she has her head screwed on right. [Slang; early 1800s]
The phrase "have one's head screwed tightly to one's shoulders" is surely less common, but its inclusion of the word tightly emphasizes the idea that the person doesn't "have a screw loose."