Using Linux (Debian, to be specific) on a laptop, I want to be able to check whether, for example, the TX or RX pairs are both connected to another machine. If, say, I've only connected RX (and am able to receive, using promiscuous mode), I want to be able to identify that TX is not connected.
There is nothing in the 802.3 standards up to 100BaseTX which would support such functionality. Of course, if you have control over both ends of the link, you could implement an echo probe which would be answered by the remote side.
For 1000BaseT things are different - there would not be such a thing as a "TX pair" or "RX pair" as all four pairs are used for transmission and reception simultaneously. As IDLE frames are continuously sent over the wire, disconnected pairs can be detected by the absence of traffic.
Additionally, some NICs may provide ways of doing cable tests, although they would be proprietary and I am not aware of any Linux implementations. This is what a Broadcom Cable Analysis dialog looks like under Windows: