Understanding the relationship between LoRa chips, chirps, symbols and bits
LoRa is a chirp-based spread-spectrum modulation. A symbol is a chirp.

To generate symbols/chirps, the modem modulates the phase of an oscillator. The number of times per second that the modem adjusts the phase is called the chip rate and defines the modulation bandwidth. Chip rate is a direct subdivision of the quartz frequency (32 MHz).

Example for 125 kHz LoRa:

125 kHz modulation bandwidth
    = 125000 chips per second
    = 8 µs per chip

modulation bandwidth &lt; occupied spectral bandwidth &lt; channel spacing (typ 200 kHz)


Basic chirps are simply a ramp from fmin to fmax (up-chirp) or fmax to fmin (down-chirp). Data-carrying chirps are chirps that are cyclically-shifted, and this cyclical shift carries the information.

The spreading factor defines two fundamental values:


the number of chips contained in each symbol is \$ 2^{SF} \$
the number of raw bits that can be encoded by that symbol is SF


The reason is that a symbol, with a length of N chips, can be cyclically shifted from 0 to N-1 positions. The "reference" position is given by the un-shifted symbols at the beginning of the LoRa frame. So this cyclical shift can carry log2(N) bits of information. If N is a power of two, the math works nicely.

Example for SF 7

A SF 7 symbol is 128 chips long
    = 1.024 ms @125kHz modulation bandwidth
    = 512 µs @250kHz modulation bandwidth
    = 256 µs @500kHz modulation bandwidth

A 128-chip long symbol can by cyclically shifted from 0 to 127 positions, and that shift
carries 7 bits of raw information:
    ~ 6.8 kbps raw @125kHz modulation bandwidth
    ~ 13.7 kbps raw @250kHz modulation bandwidth
    ~ 27.3 kbps raw @500kHz modulation bandwidth


Due to noise, this modulation/demodulation process introduces errors, and that's why an error correction code is added.
For a typical payload, 25% (CR1) or 50% (CR2) of redundancy is added before modulating chirps. In practice, the data sent by the user is also mixed to get better error correction properties.

Raw data-rate and error correction define the nominal data-rate. To get the effective maximum data-rate a device can transmit at, you have to take into account:


legal duty-cycle limit, if applicable, of the band you emit in
overhead of the LoRa preamble, header and CRC for each frame sent (significant influence when short frames are sent)
overhead of your protocol for each frame (also very important for short frames)




Edit:

I have added (in red) the boundaries of chirps so that the effect of cyclical-shifts is easier to understand. Except for a few special symbols at the end of the preamble signaling a start of frame, all chirps in a LoRa frame are the exact same length. Frequency seems to "jump around" quite a bit, but there is no discontinuity in phase that would lead to copious amounts of unwanted harmonics all around the band.