A cusp is the moment one sign changes to the next. The cusp between Libra and Scorpio happens at a precise moment, which varies from year to year. If you were born before it you would be Libra, and after it you would be Scorpio.
The moment of the Libra Scorpio cusp is not related to calendar dates. It's not as simple as saying October 22nd is always Libra and October 23rd is always Scorpio. One year the cusp may be at 11pm at night on October 22nd, and the next year it may be early in the morning on October 23rd etc.
To find out for sure which sign you are you will need to know when exactly you were born. Specifically you will need to know:
With that information an astrologer can calculate which side of the Libra Scorpio cusp you were born on, and therefore which sign you are. For example, this calculation is always performed as part of my natal readings (for one person) and relationship readings (for couples).
If you don't know your birth time then things get a little more complicated, but it is often possible to figure it out "backwards" from major life events. That's a technique called birth time rectification. I won't confuse things here by going into it, other than to say it exists for that scenario.
Another way to establish which sign you are is simply to look at the list of Libra traits compared to Scorpio traits. Libra and Scorpio have very different characteristics so it's often fairly easy to see which one you are.
In some cases however people relate almost equally to both signs. In that situation it's going to be because your sun sign is one of them, while you have another planet (most often mercury) in the other. For example you may have your sun in Libra and your mercury in Scorpio.
In that situation you probably will need an astrologer to tell you which is which, and explain how they interact. There are a large number of permutations so trying to cover them all here would be very confusing, but I automatically include your specific planets and interactions in all my natal readings (for one person) and relationship readings (for couples) mentioned above.
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