Did Texas reserve the right to secede when it joined the United States?

Yes, in the way that actually matters under the law. The popular legend that Texas signed a special deal letting it leave anytime is not quite right, and we will be honest about that. But the deeper truth is stronger than the legend: the right to resume self-government was reserved at the founding, it carries to every state equally, and Texas's own constitution states it outright.

First, the honest part about the "special deal"

You have probably heard that Texas, alone among the states, kept a written right to leave when it joined in 1845. That specific claim does not hold up, and we will not lean on it. What Texas did keep, in the annexation agreement, was the right to divide into as many as four additional states. That is real, and Chase had to ignore it to write Texas v. White. But the right to resume self-government does not depend on a Texas-only clause. It rests on something that applies to Texas and every other state alike.

Three states reserved the right to resume their powers, and Congress accepted it

When the Constitution was ratified, three states put in writing that they kept the right to take back the powers they were delegating. Virginia, ratifying on June 26, 1788, declared that the granted powers "may be resumed" by the people "whensoever the same shall be perverted to their injury or oppression." New York, on July 26, 1788, reserved that the powers of government "may be reassumed by the people, whensoever it shall become necessary to their happiness." Rhode Island said the same when it ratified in 1790. Congress seated all three without objection. That silence was acceptance. The reservation became part of the constitutional bargain the whole country joined.

The Equal Footing Doctrine carries that reservation to Texas

Here is why a reservation made by Virginia in 1788 belongs to Texas in 1845. Every state enters the union on an equal footing with the original states, with the same rights and powers, no more and no less. The Supreme Court stated it directly in Coyle v. Smith (1911): this is a union of states "equal in power, dignity and authority." You cannot have an original group of states holding a reserved right to resume self-government while every state admitted later is permanently locked in. That would not be a union of equals. Whatever Virginia, New York, and Rhode Island reserved, Texas holds in equal measure.

The Framers' own framework treats the states as sovereign parties

This is not a strained reading. It is how the men who built the system described it. Madison wrote in Federalist 39 that in ratifying, each state "is considered as a sovereign body, independent of all others, and only to be bound by its own voluntary act." A bond entered by a state's own voluntary act is the act of a sovereign party, not the surrender of a subject. The states created the federal government. They did not dissolve themselves into it.

Texas reserved the right in its own constitution, and still does

Whatever was reserved at the federal founding, Texas wrote the principle into its own fundamental law and has kept it there ever since. Article 1, Section 2 of the Texas Constitution declares that "all political power is inherent in the people" and that they have "at all times the inalienable right to alter, reform or abolish their government in such manner as they may think expedient." That language has appeared in every Texas constitution since the Republic in 1836. Texans reserved this right for themselves, in their own words, and never gave it up.

The bottom line

The cartoon version of a secret Texas-only escape clause is not the real argument, and it is not needed. The right to resume self-government was reserved at the founding, it belongs to Texas on equal footing with every other state, and Texans wrote it into their own constitution and have kept it there since 1836. The reservation is real. It is just bigger than the legend.