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Remove zest from blood oranges in wide strips with a vegetable peeler, leaving as much pith behind as possible. Set half of zest aside for serving. Slice oranges in half; squeeze juice through a fine-mesh sieve set over a measuring glass or small bowl (you should have about 1 cup juice). Discard pulp and seeds; set juice aside.
Bring ginger, cinnamon stick, cloves, sugar, remaining blood orange zest, and 1 cup water to a boil in a medium saucepan over medium heat. Cook, stirring occasionally, until sugar is dissolved and mixture is reduced by half, about 5 minutes. Let cool, then strain syrup through a clean fine-mesh sieve into a clean measuring glass; discard solids.
Stir vinegar and reserved blood orange juice into syrup. Taste shrub—it should be a well-balanced sweet-and-sour flavor; add more vinegar by the tablespoon if needed.
To serve, pour 3 Tbsp. shrub into each of 8 ice-filled glasses. Top off each with 4–5 oz. club soda and stir gently to combine. Garnish with reserved zest. Do ahead: Shrub can be made 6 weeks ahead. Pour into a 1-pint glass jar; cover and chill.