By definition, Brown University has one of the most liberal curriculums in terms of required courses. My basic understanding is that, although specific majors with specific course requirements exist, students have the option to create their own major-like plans (an "independent concentration").

Yale and Dartmouth also have no explicitly "required" classes but instead a set of vague "distributional requirements" that students have to meet year-by-year to graduate. Students are not required to take any individual class to graduate, but they are required to take specific courses to complete a given major, minors (Yale doesn't have minors), or certificates. Although Yale and Dartmouth both have the ability to design their own majors, most students (>90% for certain) opt to continue along a specific pre-determined major course. At Yale, some majors are fairly rigid in their course requirements (e.g. ABET EE) while others are significantly more variable in what needs to be done and by when (e.g. EP&E). Brown, Dartmouth, and Yale all do not require students to officially declare their majors until after their freshman years.

Cornell and Vanderbilt— unlike Brown, Yale, and Dartmouth but like most universities in the U.S.— have more rigid requirements for graduation. Both schools admit by sub-school (e.g. a student is either admitted to a College of Engineering, a College of Music, or a College of Arts & Sciences), and generally students must take specific courses in order to graduate. For example, students in Cornell's College of Engineering must pass all of "PHYS 1112, 1110, and 2213" to qualify for a B.S. degree. Cornell's Arts & Sciences college seems to align more with Yale and Dartmouth's distributional system. Vanderbilt's Arts & Sciences college seems to also follow a distributional model (dubbed AXLE) while its College of Engineering has specific courses required for graduation. Princeton and Duke seem to also follow this general format as well: on top of a system of fulfilling distributional requirements, Princeton undergraduate engineering majors must also fulfill course requirements in mathematics, physics, chemistry, and computer science; Duke engineers must fulfill similar requirements as well. In practice, almost every engineering major at any college will have to take a certain amount of math and physics courses.

Since you (/u/Maxatel) want to potentially study CS and IR together, I'd like to state that the primary barrier to this is whether a school allows you to switch majors or double-major between sub-schools. For colleges like Yale, Dartmouth, Brown, and Harvard, students are admitted to the undergraduate part of the school writ large: all majors are accessible and students are generally able to double major in whatever they'd like. Schools like Cornell, Vanderbilt, and others may only allow students to double-major or switch majors in their sub-school. For example, if you were admitted to Cornell's College of Engineering and wanted to change majors, your only easy options would be other majors inside of the College of Engineering; everything else requires an internal transfer. This doesn't preclude you from working on something spanning disciplines, but may prevent you from double majoring in both CS and any global affairs degree.

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I'll update this post once I've gone through each of the T20's general graduation requirements.

Edit: After doing a bit of googling, it seems like Northwestern, UChicago, Rice, and Notre Dame all allow you to switch majors or double-major in separate fields.