Can RX and TX be temporarily transposed on demand?
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.