Can RX and TX be temporarily transposed on demand? Only 2 arduinos can normally be linked via serial port because TX and RX need to be cross-linked, and connecting multiple TXs together would cause conflict.

RS485 allows multiple nodes connected in parallel on a 2 wire differential 'bus', using a TX ENABLE pin to allow one master to transmit while all other slaves listen. 

I suspect it should be possible to similarly connect multiple serial RXs together to give a common listening bus, and be able to use TX ENABLE on any one node at a time to transpose its RX and TX to allow its TX to transmit to the other RXs. This might be physically done using a changeover relay swapping RX with TX, but is it possible to somehow temporarily switch the TX and RX assignments on demand by software?

I doubt the actual pin assignments can be changed in-situ, but perhaps the same results are achievable with some programming wizards sleight-of-hand trickery?
On an AVR-based Arduino, you can selectively enable the transmitter and
receiver side of the UART (c.f. the datasheet of your MCU). Thus, you
could in principle just connect all the TX and RX together, and make
sure only one transmitter is enabled at a time, while all the others are
in high impedance. If you do this, I would strongly recommend you put a
current-limiting resistor in series with each TX, just in case.

For example, on an Arduino Uno
(Atmega328P),
you can use the following functions:

static inline void enable_tx()  { UCSR0B |=  _BV(TXEN0); }
static inline void disable_tx() { UCSR0B &amp;= ~_BV(TXEN0); }
static inline void enable_rx()  { UCSR0B |=  _BV(RXEN0); }
static inline void disable_rx() { UCSR0B &amp;= ~_BV(RXEN0); }


Edit: Beware that Serial.begin() enables both the transmitter and
the receiver.