You'll need the source code for the CLR to see the implementation of these methods. That's a bit hard to come by, Microsoft doesn't publish it and it is not covered by the Reference Source.
As long as the method is "old", available since .NET 2.0, then you'll have a shot at it from the SSCLI20 source code. With a non-zero risk that you will be looking at an outdated version of the code of course. But good enough to get an idea what it looks like and often still accurate.
The starting point to start searching for the code is the clr/src/vm/ecall.cpp source code file. It contains the tables that the jitter searches for internal methods. The section that's relevant for nativeCompareOrdinalEx() looks like this:
FCFuncStart(gStringFuncs)
FCDynamic("FastAllocateString", CORINFO_INTRINSIC_Illegal, ECall::FastAllocateString)
FCDynamicSig(COR_CTOR_METHOD_NAME, &gsig_IM_ArrChar_RetVoid, CORINFO_INTRINSIC_Illegal, ECall::CtorCharArrayManaged)
FCDynamicSig(COR_CTOR_METHOD_NAME, &gsig_IM_ArrChar_Int_Int_RetVoid, CORINFO_INTRINSIC_Illegal, ECall::CtorCharArrayStartLengthManaged)
FCDynamicSig(COR_CTOR_METHOD_NAME, &gsig_IM_PtrChar_RetVoid, CORINFO_INTRINSIC_Illegal, ECall::CtorCharPtrManaged)
FCDynamicSig(COR_CTOR_METHOD_NAME, &gsig_IM_PtrChar_Int_Int_RetVoid, CORINFO_INTRINSIC_Illegal, ECall::CtorCharPtrStartLengthManaged)
FCDynamicSig(COR_CTOR_METHOD_NAME, &gsig_IM_Char_Int_RetVoid, CORINFO_INTRINSIC_Illegal, ECall::CtorCharCountManaged)
FCFuncElement("nativeCompareOrdinal", COMString::FCCompareOrdinal) // <=== Here
FCFuncElement("nativeCompareOrdinalWC", COMString::FCCompareOrdinalWC)
FCIntrinsic("get_Length", COMString::Length, CORINFO_INTRINSIC_StringLength)
// etc..
}
Note how the FCFuncElement has the method name as a string and a function pointer to the C++ method that implements the internal call. Grepping the source code tree then takes you to clr/src/vm/comstring.cpp. I won't bore everybody with the C++ code, just have a look for yourself.
/*================================CompareOrdinal===============================*/
FCIMPL3(INT32, COMString::FCCompareOrdinal, StringObject* strA, StringObject* strB, CLR_BOOL bIgnoreCase) {
// Yadayada
//...
}
Searching for CaseInsensitiveCompHelper() and FastCompareStringHelperAligned() takes you to the actual implementations of the, respectively, case-insensitive and case-sensitive comparison functions in the same source code file.
The only other thing notable about this is that CLR version 4 made some changes to this mechanism. Adding lots of new internal methods and supporting an entirely different additional interop mechanism through a [DllImport] attribute for a fake DLL named "QCall". There's no good way to see source for these additions that I know of.
UPDATE: source is now available from the CoreCLR project. The table was moved from ecall.cpp to ecalllist.h, the mechanics are still the same. Do keep in mind that this is the .NETCore version of the CLR, the desktop version source is still closed-source. The two version are however likely to have much in common.