The cocktail stirrer isn't a true swizzle stick

The classic swizzle stick is quite different from what most people envision when they hear the term, and this confusion stems from an inventor named Jay Sindler. In 1934, Sindler created the modern swizzle stick after struggling to extract an olive from his martini.
In his patent, Sindler claimed that his invention—a small wooden spear featuring a space for advertising the bar or restaurant's name—would put an end to the "boorish antics that offend those accustomed to polite social customs" associated with fishing an olive out of a glass (via Punch). By the 1950s, which marked the peak of the plastic swizzle stick (also known as a drink stirrer), this vibrant and creatively designed version had largely replaced the original Caribbean swizzle stick.
The swizzle stick versus the cocktail shaker

Bartenders in the Caribbean have upheld the tradition of swizzling drinks with a classic wooden stick. This technique, along with the delightful beverages it creates, is increasingly influencing cocktail culture today. So, when is it appropriate to opt for a traditional swizzle stick instead of a cocktail shaker? For true swizzling enthusiasts, only sour-style drinks typically made with rum and crushed ice qualify as authentic swizzles. Consequently, these are the only cocktails that necessitate this mixing method. A prime example of a classic swizzle is the rum swizzle, Bermuda's national drink, which blends orange, lemon, and pineapple juices with rum, grenadine, and bitters, all swirled to frothy perfection using a wooden swizzle stick.
Some bartenders argue that any cocktail typically shaken can also be swizzled, as both methods achieve similar results. Both shaking and swizzling mix the ingredients, dilute the alcohol, chill the drink quickly, and create a frothy texture. Additionally, swizzling simplifies the process since it’s done directly in the serving glass. Let’s not overlook the modern plastic swizzle stick, either. Feel free to incorporate one into your nostalgic drinks for a stylish touch, like a retro 7 and 7 instead of a Jack and Coke. The Seagram's 7 Crown Whiskey paired with 7 Up could certainly benefit from a good stir.
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