You shouldn't twist a tick at all. See this website sponsored by the National Institutes of Health for proper tick removal techniques:
If a tick is attached to you, follow
these steps to remove it:
Grasp the tick close to its head or mouth with tweezers. Do not use your
bare fingers. If needed, use a tissue
or paper towel.
Pull it straight out with a slow and steady motion. Avoid squeezing or
crushing the tick. Be careful not to
leave the head embedded in the skin.
Clean the area thoroughly with soap and water. Also wash your hands
Save the tick in a jar and watch carefully for the next week or two for
signs of Lyme disease.
If all parts of the tick cannot be removed, get medical help. Bring the
tick in the jar to your doctor's
• Do NOT try to burn the tick with a
match or other hot object.
• Do NOT
twist the tick when pulling it out.
• Do NOT try to kill, smother, or
lubricate the tick with oil, alcohol,
vaseline, or similar material
It is possible to remove a tick and accidentally leave the mouth parts in, but the mouth parts traveling to the bloodstream is not the concern. The concern focuses around the increased chances of infection (both from the wound, and because the mouth parts may still secrete saliva, which might transmit disease).