On I-20, between Midland and Stanton, TX, the EB Frontage Road is on the south side of the tracks, while the freeway and the WB frontage road are on the north side of the tracks.
Old Maps show Original US 80 being on the north side of the tracks, and that it became a divided highway before becoming I-20. Looks like they did this because it was easier cheaper to leave the original divided highway alone.
This condition extends west into Midland on BUS I-20, and interestingly, the frontage road on the south side of the tracks has typical TxDOT seal coat, but has no TxDOT signs. Stop signs along it are on green U-channel posts. Looks like its not currently state maintained.
https://www.google.com/maps/place/FM+1053,+Imperial,+TX/@32.0420115,-101.8736158,23105m/data=!3m1!1e3!4m5!3m4!1s0x86f07f081beb1887:0xabd2136b48547e8b!8m2!3d31.2682343!4d-102.6972182 (https://www.google.com/maps/place/FM+1053,+Imperial,+TX/@32.0420115,-101.8736158,23105m/data=!3m1!1e3!4m5!3m4!1s0x86f07f081beb1887:0xabd2136b48547e8b!8m2!3d31.2682343!4d-102.6972182)
There was a short section like this on I-35E in Lewisville, at the Garden Ridge Exit, but the frontage road was moved to the west side of the tracks when the highway was recently rebuilt. Oddly, in this instance, the RR and Highway were built on this new location at the same time in the early 50's as Lake Lewisville dam was being constructed.
https://www.google.com/maps/place/FM+1053,+Imperial,+TX/@33.0822174,-97.0256784,714m/data=!3m1!1e3!4m5!3m4!1s0x86f07f081beb1887:0xabd2136b48547e8b!8m2!3d31.2682343!4d-102.6972182 (https://www.google.com/maps/place/FM+1053,+Imperial,+TX/@33.0822174,-97.0256784,714m/data=!3m1!1e3!4m5!3m4!1s0x86f07f081beb1887:0xabd2136b48547e8b!8m2!3d31.2682343!4d-102.6972182)
When I saw the topic title I thought of the I-20 example you mentioned.
In Midland, TxDOT's Statewide planning map has the south frontage road labelled as County Road 87. The roadway inventory overlay does not highlight it, as it does with state-maintained frontage roads. That road is given a route ID that is not related to BI-20. The north frontage road is highlighted, and is called BI0020E-XG, with is consistent with the nomenclature for state frontage roads. This also suggests that the south frontage road is not state maintained.
Edit: I'm sure the OP has this link, but new visitors may be interested in the map and not know where it is, so here's a link
http://www.txdot.gov/apps/statewide_mapping/StatewidePlanningMap.html (http://www.txdot.gov/apps/statewide_mapping/StatewidePlanningMap.html)
A minor tangent: I really like how Impact! Chemical Co's sign still has Hwy 80 as their address:
https://www.google.com/maps/@32.0768223,-101.9180017,0a,75y,58.73h,90.37t/data=!3m4!1e1!3m2!1shIW4eoPPVmZdQfty0krG-A!2e0
I think I saw part of I-27 this way when once looking at GSV.