Which CMOS logic families can safely be used to construct linear circuits?

All logic families like to use buffered inverters, because those are more reliable and use less power in digital applications. However, unbuffered inverters are useful to build crystal oscillators, so they exist in many families; search for 74xx1GU04.

A 5 V-tolerant I/O has no ESD protection diode to VCC, so it tends to have less capacitance, and distorts the signal less if it exceeds VCC.

TTL-compatible inputs have a lower switching threshold, so they are no longer symmetric between VCC and ground.

Unbuffered gates are meant to be used in linear circuits; buffered gates are unlikely to work at all.

Another useful application note: Understanding (un)buffered CD4xxx characteristics.

You have to remember logic gates like invertors are really just simple analog circuits, comparators, tailored to work well with an analog input signal that basically has two stable states, high and low.

As such, just like you can use op-amps as logic devices, simple logic devices can also be used in an analog role.

Invertors in particular fill this role nicely, since what you really have is a simple comparator/op-amp with the negative pin exposed as the input and the positive pin basically "connected" to half rail. (Or some other point for TTL etc.) Because they expose the negative pin, you can use negative feedback loops in the same way you do with op-amps. Non inverting logic is less useful.

How well they work in an analog role is of course dependent on the nature of the particular gate. Older devices are very simple matched transistors, the buffered variety have more internals that make them less linear.

Logic devices do however have a tendency to open-circuit, or worse, shoot-though, when the signal is between logic levels so using them as simple amplifiers for low frequency signals is not a great idea.

However using them as part of a delay circuit, or as the driver in an oscillator, they work well especially if the gate is a Schmitt Trigger with it's built in hysteresis.

I belatedly wanted to add a few points that weren't elaborated by others.

While it is customary to use unbuffered gates as linear amplifiers, there are a few drawbacks that must be borne in mind.

Perhaps most importantly, the parameters are poorly specified. While an amplifier datasheet has a lot of information about the amplifier properties, you will usually find very little such information in the datasheet of a logic device. Furthermore, there are bound to be large tolerances and variability over operating conditions (operating voltage, temperature, ...). Hence, you may only want to employ those devices in circuits that can tolerate such large variations.

Unbuffered inverters are available in various different CMOS logic families, starting with the old 4000 series on the slow end, up to the quite fast LVC range. Their properties differ markedly. You want to have a close look at power consumption in particular, since power draw tends to be at maximum when the input voltage is mid-range between high and low, where both transistors conduct simultaneously. This is going to be very dependent on operating voltage, too. It gets worse the faster and higher output drive the logic family is. This is why the 4000 series is fairly benign, whereas LVC type logic is much harder to deal with.

Depending on the logic family, there may also be a specified maximum signal rise/fall time, which indicates that the input level isn't supposed to stay between high and low for long. If you violate this, you don't just get high power consumption, you may also run into stability problems. It might even affect the reliability of the circuit, due to the heat generated in a fairly small pair of transistors. TI application note SCBA004 has more to say about this.

Bottomline is: You may use those devices for linear applications if you are aware of the serious limitations. Their low price may be attractive, but the drawbacks that come with the simple circuit are substantial.