Can an LG G2 phone work on both GSM and CDMA?

Short Answer: Yes, but probably not

Long Answer: There were 8 Versions of the LG G2 released during September 2013, below I'll have the versions listed so you can reference.

Good News: of those 8, 2 support both GSM & CDMA, Verizon & Sprint

Bad News: Sprint's version supports more bands, BUT Verizon & Sprint supports different LTE Bands. What's worse is that neither of the LTE band sets are supported by most other GSM/CDMA carriers, so you'll be stuck on 3G. Lastly though, good luck on getting either unlocked.

~LG-D802(Global, GSM)

GSM 850/900/1800/1900

LTE 800/900/1800/2100/2600

~LG-D803(Canada, GSM)

GSM 850/900/1800/1900

LTE 700, AWS, 2600

~LG-DB05(South America, GSM)

GSM 850/900/1800/1900

LTE 700(17)/1700(4)/2600(7)

~LG-D800( U.S. AT&T, GSM)

GSM 850/900/1800/1900

LTE 1900(2)/1700(4)/850(5)/700(17)

~LG-D801( U.S. T-Mo, GSM)

GSM 850/900/1800/1900

LTE 1900(2)/1700(4)/700(17)

~LS980( U.S. Sprint, CDMA/GSM)

GSM 850/900/1800/1900

CDMA 800/850/1900

LTE 800 (26) /1900 (25) /2500 (41)

~VS980( U.S. Verizon, CDMA/GSM)

GSM 850/900/1800/1900

CDMA 850/1900

LTE 1700(4)/700(13)

~L-01F(Japan, Carrier Specific GSM)

GSM 850/900/1800/1900

LTE 2100(1)/800(19)/1500(21)