Intercultural Development Inventory (IDI)
Cultural competence has to do with one's awareness of one's own cultural "baggage" (identity) and the awareness and ability to interact effectively with individuals who have a distinct cultural norms. The intercultural development inventory measures cultural competency as a continuum of five scales ("stage"?):

1. Denial: Limited experience and underdstanding of differences;
2. Polarization: "Us vs. Them" mindset; judges differences;
3. Minimization: Emphasizes similarities, equality, inclusiveness;
4. Acceptance: recognizes and appreciates cultural differences;
5. Adaptation: Able to bridge deep culture differences across diverse communities;

The research behind the tool has been published in the early 2000 by M. Hammer and colleagues. Here is one of their citations: Measuring intercultural sensitivity: The intercultural development inventory (https://doi.org/10.1016/S0147-1767(03)00032-4). Additional information is available on the site that makes the scale commercially available: https://idiinventory.com.

Measuring global cultural knowledge and awareness aptitude
The Global Competence Aptitude Assessment is a commercial instrument for providing information about a person's level of global competence because it generates a comprehensive and detailed report on global skills, attitudes and knowledge. The web site has a page on Presentations and Peer-reviewed Research Publications.

Cross-Cultural Adaptability (CCAI) Inventory
The CCAI instrument is a commercial product to help businesses identify their current strengths and weaknesses of individuals within four critical skill areas important for effective cross-cultural communication and interaction. It is claimed that the instrument helps provide insight into the ability to adapt to new situations, interact with people different from oneself, tolerate ambiguity, and maintain a sense of self in new or different surroundings. The tool has been used to measure the impact of study abroad programs on undergraduate students; see for example Shaftel et al. 2007 in Text book and readings page. Although the company web site provides a peer-reviewed and other articles in support of CCAI, a 2006 article questioned it validity: Davis, S. L. and S. J. Finney. 2006. A Factor Analytic Study of the Cross-Cultural Adaptability Inventory.

Cultural Intelligence Scale
Soon Ang, Linn Van Dyne, Christine Koh, K. Yee Ng, Klaus J. Templer, Cheryl Tay and N. Anand Chandrasekar. 2007. Cultural Intelligence: Its Measurement and Effects on Cultural Judgment and Decision Making, Cultural Adaptation and Task Performance. Management and Organization Review 3(3): 335–371. The Cultural Intelligences (CIs) Scale measures the capability to function effectively in culturally diverse settings. This research developed and tested a model that posited differential relationships between the four cultural intelligence dimensions (metacognitive, cognitive, motivational and behavioural) and three intercultural effectiveness outcomes (cultural judgment and decision making, cultural adaptation and task performance in culturally diverse settings). The results demonstrate a consistent pattern of relationships where metacognitive and cognitive CIs predicted cultural judgment and decision making; motivational and behavioural CIs predicted cultural adaptation; and metacognitive and behavioural CIs predicted task performance.

Measuring Cultural and Ethnic Identities (see more here:...)
Phinney, J. S. (1992). The Multigroup Ethnic Identity Measure: A new scale for use with adolescents and young adults from diverse groups. Journal of Adolescent Research, 7, 156–176. This paper presents a questionnaire measure of ethnic identity based on elements of ethnic identity that are common across groups, so that it can be used across all groups. Reliability, assessed by Cronbach's alpha, was .81 for the high school sample (n=417), and .90 for the college sample (n=136). Whites scored lower in ethnic identity than members of the three minority groups.

Ponterotto, J. H. 2003. The Multigroup Ethnic Identity Measure (MEIM): Psychometric Review and Further Validity Testing. Educational and psychological measurement 63(3):502 -515.

Avery, Derek R. 2007. Assessing the Multigroup Ethnic Identity Measure for measurement equivalence across racial and ethnic groups. Educational and psychological measurement 67(5): 877-888.

Awareness of Racial Privileges, Institutional Discrimination and Blatant Racial Issues: Color-Blind Racial Attitudes Scale
Neville, H. A., Lilly, R. L., Duran, C., Lee, R. M., & Browne, L. (2000). Construction and initial validation of the color-blind racial attitudes scale (CoBRAS). Journal of Counseling Psychology, 47, 59–70. The CoBRAS is designed to assess “cognitive dimensions of color-blind racial attitudes.” It consists of 20 items to assess color-blind racial attitudes with a 6-point Likert scale of 1 (strongly disagree) to 6 (strongly agree). Higher scores show greater levels of blindness or unawareness. The CoBRAS includes blindness to three areas: Racial Privileges, Institutional Discrimination, and Blatant Racial Issues. The Racial Privilege subscale measures blindness to the existence of White privileges. The Institutional Discrimination subscale measures limited awareness of the implications of institutional discrimination and exclusion. The Blatant Racial Issues subscale measures unawareness of general and pervasive racial discrimination. Neville et al. (2000) reported that the coefficient alpha for the total scale was .91.

Gamst, G. C., Liang, C. T., & Der-Karabetian, A. (2011). Ch 7. Racism- and prejudice-related measures. In Handbook of multicultural measures. (pp. 251-335). Thousand Oaks, CA: SAGE Publications, Inc. doi: 10.4135/9781412996754.n7 (download Measure-Racial-Prejudices.pdf).

Measuring Awareness and Aptitude in Counseling Across Cultures
Ponterotto, J. G., Gretchen, D., Utsey, S. O., Riger, B. P., & Austin, R. (2002). A revision of the multicultural counseling awareness scale. Journal of Multicultural Counseling and Development, 30, 153–181. The Multicultural Counseling Knowledge and Awareness Scale (MCKAS) is made of 32 items to assess self reported multicultural counseling knowledge and awareness. The MCKAS uses a 7-point Likert scale of 1 to 7, higher scores indicating greater perceived knowledge and awareness of multicultural counseling issues. Scale scores are calculated by adding items in each subscale, higher scores showing greater self-assessed competence in respective areas