“the 10th Amendment was cited by the seceeding States in support of their formal severance of ties.”
No, the 10th Amendment was not cited by any of the 11 seceding states in their Ordinances of Secession.
Five of the seceding states wrote documents laying out the reason and justification for their decisions to secede from the Union. Not one of those documents cites justification under the 10 Amendment to the Constitution. In two of those documents the Constitutional article cited is Article IV, Section 2 clause 3 (aka the fugitive slave clause). One state cites Article IV section 2 clause 2. One state cites Article 7. No where in the primary documentation of secession is Amendment 10 of the Constitution mentioned.
I disagree - here are a few examples (pulled up from memory):
By this Constitution, certain duties were imposed upon the several States, and the exercise of certain of their powers was restrained, which necessarily implied their continued existence as sovereign States. But to remove all doubt, an amendment was added, which declared that the powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States, respectively, or to the people.
South Carolina - Declaration of the Immediate Causes Which Induce and Justify the Secession of South Carolina from the Federal Union, 1860
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Senators, the Constitution is a compact. It contains all our obligations and the duties of the federal government… I say that the Constitution is the whole compact. All the obligations, all the chains that fetter the limbs of my people, are nominated in the bond, and they wisely excluded any conclusion against them, by declaring that “the powers not granted by the Constitution to the United States, or forbidden by it to the States, belonged to the States respectively or the people.”
Robert Augustus Toombs of Georgia – Remarks upon resigning from the US Senate, 1861
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The Constitution of 1787, having, however, omitted the clause already recited from the Articles of Confederation, which provided in explicit terms that each State retained its sovereignty and independence, some alarm was felt in the States, when invited to ratify the Constitution, lest this omission should be construed into an abandonment of their cherished principle, and they refused to be satisfied until amendments were added to the Constitution placing beyond any pretense of doubt the reservation by the States of all their sovereign rights and powers not expressly delegated to the United States by the Constitution. Strange, indeed, must it appear to the impartial observer, but it is none the less true that all these carefully worded clauses proved unavailing to prevent the rise and growth in the Northern States of a political school which has persistently claimed that the government thus formed was not a compact between States, but was in effect a national government, set up above and over the States.
Jefferson Davis - Message to Congress, April 29, 1861
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Perhaps you were doing a quick search, using terms such as "Tenth Amendment" or "10th Amendment". What was commonly cited was the language (or paraphrased language) of the amendment itself...
The 10th Amendment applies to all non-prohibitied powers not specifically enumerated to and for the general government. The power of secession is one of those un-enumerated powers reserved to the states and the people, as mentioned in the South Carolina declaration of Secession:
"The people of the State of South Carolina, in Convention assembled, on the 26th day of April, A.D., 1852, declared that the frequent violations of the Constitution of the United States, by the Federal Government, and its encroachments upon the reserved rights of the States, fully justified this State in then withdrawing from the Federal Union; but in deference to the opinions and wishes of the other slaveholding States, she forbore at that time to exercise this right. Since that time, these encroachments have continued to increase, and further forbearance ceases to be a virtue…
"By this Constitution, certain duties were imposed upon the several States, and the exercise of certain of their powers was restrained, which necessarily implied their continued existence as sovereign States. But to remove all doubt, an amendment was added, which declared that the powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States, respectively, or to the people. On the 23d May, 1788, South Carolina, by a Convention of her People, passed an Ordinance assenting to this Constitution, and afterwards altered her own Constitution, to conform herself to the obligations she had undertaken.
"Thus was established, by compact between the States, a Government with definite objects and powers, limited to the express words of the grant. This limitation left the whole remaining mass of power subject to the clause reserving it to the States or to the people, and rendered unnecessary any specification of reserved rights."
["Declaration of the Immediate Causes Which Induce and Justify the Secession of South Carolina from the Federal Union." Avalon Project, Dec 24, 1860]
Therefore, South Carolina believed the 10th Amendment applied, and specifically expressed that belief.
Mr. Kalamata