Place about 10 ice cubes in a plastic bag with a zipper-like closure. With a mallet, a rolling pin or the handle of a large knife, crack the ice into coarse pieces. Transfer them to the bowl of a food processor. Pulse until ice is finely shaved, but not powdery. Transfer it to a chilled bowl. Working with your hands, gently gather ice shavings and shape into a 2 to 4-inch ball, depending on size of serving glass. Do not pack the ice too tightly. Place ball in a chilled glass.
Fill a cocktail shaker one-third of the way up with crushed ice. Add vodka, orange liqueur, lime juice and cranberry juice. Shake well and strain liquid over the snowball. Garnish with the lemon or lime twist and serve with a small cocktail straw.