Commentators are puzzled by an apparent anomaly of Lawrence: its strong reliance on liberty--which, under the scrutiny regime, enjoys only minimum scrutiny--and its use of minimum scrutiny jargon to strike down the sodomy law. The anomaly is resolved when we understand the case which both discussed liberty and established the scrutiny regime: West Coast Hotel v. Parrish, 300 US 379 (1937). In that case, the state argued only that its minimum wage law was "a reasonable exercise of the police power of the state." Id., at 389.
However, the Court, in sustaining the law, said that the law restrained "an absolute and uncontrollable liberty." Id., at 391. The Court here assumed that absoluteness and uncontrollability were aspects of liberty, and that such a liberty had ever been the law. Nevertheless, this was interpreted as meaning that liberty should only be regarded as an interest, subject to minimum scrutiny. The problem with Lawrence was that the Court did not go back to examine the history of its own use of the word liberty: it simply trotted out both liberty and the scrutiny regime. West Coast Hotel is not cited at all in Lawrence.
Lawrence is probably best seen as an implicitly overruling of the scrutiny regime itself because liberty is removed from it, and the regime has no other foundation besides a restriction of liberty. On the other hand, the Court does not say how liberty is now adjudicated in the Constitution. The Court said that the sodomy law "furthers no legitimate state interest." Lawrence at 578. However, that is not the part of the rational basis test it is usually taken to be. The rational basis test says that the government policy must be rationally related to a legitimate state interest. Intermediate scrutiny says that the government policy must substantially further an important state interest.
To see that the Court has abandoned the scrutiny regime we need only ask, with regard to the Lawrence test, how much further? to realize that that question is not relevant if the state interest need only be "legitimate." We have reached the end of the scrutiny regime. The Court is not using the scrutiny regime AT ALL in adjudicating liberty.
The Court's Lawrence test means what it says: government policy affecting liberty must "further a legitimate state interest." What that means, in turn, depends on what, in fact, is liberty. That is the investigation to which the Court is telling us to turn our attention.